Aapti by Amar

8/12/2019 Aapti by Amar

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Important Formulas

Surds and Indices

1)

2)

3)

4)

5)

6)

7)

8)

9)

10)

Laws of Indices:

am= am- n

an

(ab)n= anbn

a n=

b

a0= 1

Some Basic Formulae:

i. (a+ b)(a- b) = (a2- b2)

ii. (a+ b)2= (a2+ b2+ 2ab)

iii. (a- b)2= (a2+ b2- 2ab)

iv. (a+ b+ c)2= a2+ b2+ c2+2(ab+ bc+ ca)

v. (a3+ b3) = (a+ b)(a2- ab+ b2)

vi. (a3- b3) = (a- b)(a2+ ab+ b2)

vii. (a3

+ b3

+ c3

- 3abc) = (a+ b+ c)(a2+ b2+ c2- ab- bc- ac)

viii. When a+ b+ c= 0,then a3+ b3+ c3= 3abc.


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Average

1. Average:

Average = Su !" !b#ervat$!n#%uber !" !b#ervat$!n#

2. Average Speed:

Su&&!#e a an '!ver# a 'erta$n $#tan'e atx&h an an e*ua$#tan'e aty&h.

hen, the average #&ee ru$ng the h!e !urne/ $#

2xy

&h.x+

y

Percentages t! e&re## ab a# &er'entage---- (a/b) !""

t! e&re## a &er'entage !" b------ (a/!"") b

" the &r$'e !" an $te $n'rea#e# b/ A, then the reu't$!n $n'!n#u&t$!n #! a# n!t t! $n'rea#e the e&en$ture $#-------- A / (!"" # A) !"" $

" the &r$'e !" an $te e'rea#e# b/ A, then the $n'rea#e $n '!n#u&t$!n#! a# n!t t! e'rea#e the e&en$ture $#------ A / (!"" % A) !"" $

1. &oncept of Percentage:

/ a 'erta$n percent, e ean that an/ hunreth#.

hu#,x&er'ent ean#xhunreth#, r$tten a#x.

! e&re## a a# a &er'ent We have, a = a 100.b b b

hu#,1

=1

100

= 25.4 4

2. Percentage Increase/'ecrease:


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" the &r$'e !" a '!!$t/ $n'rea#e# b/ , then the reu't$!n$n '!n#u&t$!n #! a# n!t t! $n'rea#e the e&en$ture $#

100

(100 + )

" the &r$'e !" a '!!$t/ e'rea#e# b/ , then the $n'rea#e $n'!n#u&t$!n #! a# n!t t! e'rea#e the e&en$ture $#

100

(100 - )

Prot And Loss Prot Selling Price % &ost Price Loss &ost Price % Selling Price Prot Percentage (Prot / &ost Price) !"" Loss Percentage (Loss / &ost Price) !""

1. a$n = (S..) - (..)

2. :!## = (..) - (S..)

3. :!## !r ga$n $# aa/# re'!ne !n ..

4. a$n er'entage (a$n )

a$n =a$n 100

..

5. :!## er'entage (:!## )

:!## =:!## 100

..6. Se$ng r$'e (S..)

S =(100 + a$n )

.100

7. Se$ng r$'e (S..)

S =(100 - :!## )

..100

8. !#t r$'e (..) .. = 100 S..


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(100 + a$n )

9. !#t r$'e (..)

.. =100

S..(100 - :!## )

10." an art$'e $# #! at a ga$n !" #a/ 35, then S.. = 135 !" ..

11." an art$'e $# #! at a !## !" #a/, 35 then S.. = 65 !" ..

12.When a &er#!n #e# t! #$$ar $te#, !ne at a ga$n !" #a/x,an the !ther at a !## !"x, then the #eer aa/# $n'ur# a !##g$ven b/

:!## =!!n :!## an a$n 2

= x 2

.

10 1013." a traer &r!"e##e# t! #e h$# g!!# at '!#t &r$'e, but u#e# "a#ee$ght#, then

a$n =;rr!r

100.(rue


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htt&.'at4ba.'!ath-e-b!!&r!t-!## [1]

*rue 'iscount

Su&&!#e a an ha# t! &a/ #. 156 a"ter 4 /ear# an the rate !"$ntere#t $# 14 &er annu. ear/, #. 100 at 14 $ a!unt t! .156 $n 4 /ear#. S!, the &a/ent !" #. n! $ 'ear !> the ebt !" #.156 ue 4 /ear# hen'e. We #a/ that

Su ue = #. 156 ue 4 /ear# hen'e?

re#ent W!rth (.W.) = #. 100?

rue @$#'!unt (.@.) = #. (156 - 100) = #. 56 = (Su ue) - (.W.)

We ene *+'+ Interest on P+,+- Amount (P+,+) # (*+'+)

ntere#t $# re'!ne !n .W. an true $#'!unt $# re'!ne !n thea!unt.


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I.P0*A1* F0.2LA3

:et rate = &er annu an $e = /ear#. hen,

1. .W. =

100 A!unt

=

100 .@.

100 + ( )

2. .@. =(.W.)

=A!unt

100 100 + ( )

3. Su =(S..) (.@.)

(S..) - (.@.)

4. (S..) - (.@.) = S.. !n .@.

5. When the #u $# &ut at '!&!un $ntere#t, then .W.=

A!unt

1+

100

Simple and &ompound Interest

1) !r ntere#t !&!une Annua/ *otal Amount P(!#(0/!"")) n

2) !r ntere#t !&!une Ba" Cear/ *otal Amount P(!#(0/4"")) 4n

3) !r ntere#t !&!une Duarter/ *otal Amount P(!#(0/5""))5n

4) S$&e ntere#t = (r$n'$&a E ate E $e) 100SI P0*/!""Where = 'a&$ta a!unt, = annua rate, = t$e &er$! $n /ear#

&ompound Interest

1. :et r$n'$&a = , ate = &er annu, $e = n/ear#.

2. ,6en interest is compound Annuall7:

A!unt = 1 + n100

3. ,6en interest is compounded 8alf%7earl7:

A!unt = 1 +(2) 2n

100


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4. ,6en interest is compounded 9uarterl7:

A!unt = 1 +(4) 4n

100

5. ,6en interest is compounded Annuall7 but time is in

fraction sa7 ; 7ears+

A!unt = 1 + 3

1 +

100 100

6. ,6en 0ates are di


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the t$e# taen b/ then t! '!ver the #ae $#tan'e $#1

1

!r b a.a b

5. Su&&!#e a an '!ver# a 'erta$n $#tan'e atxhr an an e*ua$#tan'e atyhr. hen,

the average #&ee ur$ng the h!e !urne/ $#

2xy

hr.x+y

*rain Problems

1. =m/6r to m/s conversion:

ahr = a5

#.182. m/s to =m/6r conversion:

a# = a18

hr.5

3. $e taen b/ a tra$n !" ength letre# t! &a## a &!e !r #tan$ngan !r a #$gna &!#t $# e*ua t! the t$e taen b/ the tra$n t!'!ver letre#.

4. $e taen b/ a tra$n !" ength letre# t! &a## a #tat$!ner/

!be't !" ength betre# $# the t$e taen b/ the tra$n t! '!ver(l+ b) etre#.

5. Su&&!#e t! tra$n# !r t! !be't# b!$e# are !v$ng $n the #ae$re't$!n at u# an v#, here uF v, then the$r reat$ve#&ee $# = (u- v) #.

6. Su&&!#e t! tra$n# !r t! !be't# b!$e# are !v$ng $n !&&!#$te$re't$!n# at u# an v#, then the$r reat$ve #&ee $# =(u+ v) #.

7. " t! tra$n# !" ength aetre# an betre# are !v$ng $n!&&!#$te $re't$!n# atu# an v#, then

he t$e taen b/ the tra$n# t! 'r!## ea'h !ther =(a+ b)

#e'.(u+ v)


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8. " t! tra$n# !" ength aetre# an betre# are !v$ng $n the#ae $re't$!n atu# an v#, then

he t$e taen b/ the "a#ter tra$n t! 'r!## the #!er tra$n=

(a+b) #e'

(u- v)

9. " t! tra$n# (!r b!$e#) #tart at the #ae t$e "r! &!$nt# A an t!ar# ea'h !ther an a"ter 'r!##$ng the/ tae aan b#e' $nrea'h$ng an A re#&e't$ve/, then(AG# #&ee) (G# #&ee) = (b a)

Boats And Streams

:et b = b!at #&ee, # = #trea #&ee, = $#tan'e an t = t$eH?&F

1. Smoot6 water e>uation :% b d / t2. 2pstream e>uation - b # s d/t3. 'ownstream speed :% b % s d/t

*ime and ,or=1. ,or= from 'a7s:

" A 'an ! a &$e'e !" !r $n na/#, then AG# 1 a/G# !r =1

n2. 'a7s from ,or=:

" AG# 1 a/G# !r =1

, then A 'an n$#h the !r $n na/#.n

3. 0atio:" A $# thr$'e a# g!! a !ran a# , thenat$! !" !r !ne b/ A an = 3 1.at$! !" t$e# taen b/ A an t! n$#h a !r = 1 3.

1! 0! 14 04 ,Bere % = nuber !" !r$ng en? = rate !" !r !ne &er an ? @= a/# ?W = t!ta !r !ne.

$&e# an $#tern


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1. Inlet:

A &$&e '!nne'te $th a tan !r a '$#tern !r a re#erv!$r, that #$t, $# n!n a# an $net.

utlet:

A &$&e '!nne'te $th a tan !r '$#tern !r re#erv!$r, e&t/$ng $t,$# n!n a# an !utet.

2. " a &$&e 'an a tan $nxh!ur#, then

&art e $n 1 h!ur =1

.x

3. " a &$&e 'an e&t/ a tan $nyh!ur#, then

&art e&t$e $n 1 h!ur =1

.y

4. " a &$&e 'an a tan $nxh!ur# an an!ther &$&e 'an e&t/ the"u tan $nyh!ur# (hereyFx), then !n !&en$ng b!th the &$&e#,then

the net &art e $n 1 h!ur =1

-1

.x y

5. " a &$&e 'an a tan $nxh!ur# an an!ther &$&e 'an e&t/ the"u tan $nyh!ur# (hereyFx), then !n !&en$ng b!th the &$&e#,then

the net &art e&t$e $n 1 h!ur =1

-1

.y x

0atio and Proportion

1. 0atio:


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he rat$! !" t! *uant$t$e# aan b$n the #ae un$t#, $# the

"ra't$!n an e r$te $t a# a b.

n the rat$! a b, e 'a aa# the r#t ter !r antecedentan b,

the #e'!n ter !r conse>uent.

;g. he rat$! 5 9 re&re#ent#5

$th ante'eent = 5, '!n#e*uent = 9.9

0ule:he ut$&$'at$!n !r $v$#$!n !" ea'h ter !" a rat$! b/ the#ae n!n-Ier! nuber !e# n!t a>e't the rat$!.

;g. 4 5 = 8 10 = 12 15. A#!, 4 6 = 2 3.

2.Proportion:

he e*ua$t/ !" t! rat$!# $# 'ae &r!&!rt$!n.

" a b= c d, e r$te a b::c dan e #a/ that a, b, c, dare$n &r!&!rt$!n.

Bere aan dare 'ae etremes, h$e ban care 'ae meanterms.

r!u't !" ean# = r!u't !" etree#.

hu#, a b::c d (b c) = (a d).

3. Fourt6 Proportional:

" a b= c d, then d$# 'ae the "!urth &r!&!rt$!na t! a, b, c.

*6ird Proportional:

a b= c d, then c$# 'ae the th$r &r!&!rt$!n t! aan b.

.ean Proportional:

Jean &r!&!rt$!na beteen aan b$# ab.

4. &omparison of 0atios:

We #a/ that (a b) F (c d) a F c .


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b d

5. &ompounded 0atio:

6. he '!&!une rat$! !" the rat$!# (a b), (c d), (e f) $#

(ace bdf).

7. 'uplicate 0atios:

@u&$'ate rat$! !" (a b) $# (a2 b2).

Sub-u&$'ate rat$! !" (a b) $# (a b).

r$&$'ate rat$! !" (a b) $# (a3 b3).

Sub-tr$&$'ate rat$! !" (a b) $# (a13

b13

).

" a = c , then

a+b =

c+d . K'!&!nen! an $v$en!L

b d a- b c- d

8. ?ariations:

We #a/ thatx$# $re't/ &r!&!rt$!na t!y, $"x= ky"!r #!e'!n#tant kan e r$te,x y.

We #a/ thatx$# $nver#e/ &r!&!rt$!na t!y, $"xy= "!r #!e'!n#tant k and

e r$te,x1

.y

r!be# !n Age#

1. dd 'a7s:

We are #u&&!#e t! n the a/ !" the ee !n a g$ven ate.

!r th$#, e u#e the '!n'e&t !" G! a/#G.


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n a g$ven &er$!, the nuber !" a/# !re than the '!&eteee# are 'aeodd da7s.

2. Leap @ear:

($). ;ver/ /ear $v$#$be b/ 4 $# a ea& /ear, $" $t $# n!t a 'entur/.

($$). ;ver/ 4th'entur/ $# a ea& /ear an n! !ther 'entur/ $# a ea&/ear.

%!te A leap 7ear 6as ; da7s+

3amples:

$. ;a'h !" the /ear# 1948, 2004, 1676 et'. $# a ea& /ear.

$$. ;a'h !" the /ear# 400, 800, 1200, 1600, 2000 et'. $# a ea&/ear.

$$$. %!ne !" the /ear# 2001, 2002, 2003, 2005, 1800, 2100 $# aea& /ear.

3. rdinar7 @ear:

he /ear h$'h $# n!t a ea& /ear $# 'ae an ordinar7 7ears. An!r$nar/ /ear ha# 365 a/#.

4. &ounting of dd 'a7s:

1. 1 !r$nar/ /ear = 365 a/# = (52 ee# + 1 a/.)

1 !r$nar/ /ear ha# 1 ! a/.

2. 1 ea& /ear = 366 a/# = (52 ee# + 2 a/#)

1 ea& /ear ha# 2 ! a/#.

3. 100 /ear# = 76 !r$nar/ /ear# + 24 ea& /ear#

= (76 1 + 24 2) ! a/# = 124 ! a/#.

= (17 ee# + a/#) 5 ! a/#.

%uber !" ! a/# $n 100 /ear# = 5.


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%uber !" ! a/# $n 200 /ear# = (5 2) 3 ! a/#.

%uber !" ! a/# $n 300 /ear# = (5 3) 1 ! a/.

%uber !" ! a/# $n 400 /ear# = (5 4 + 1) 0 ! a/.

S$$ar/, ea'h !ne !" 800 /ear#, 1200 /ear#, 1600 /ear#,2000 /ear# et'. ha# 0 ! a/#.

2 'a7 of t6e ,ee= 0elated to dd 'a7s:

%!. !" a/# 0 1 2 3 4 5 6

@a/ Sun. J!n. ue#. We. hur#. r$. Sat.

Area

. 1. Area !" a re'tange = (:ength reath).

:ength =Area

an reath =Area

.reath :ength

. 2. er$eter !" a re'tange = 2(:ength + reath).

. Area !" a #*uare = (#$e)2= ($ag!na)2.


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3. Area !" an e*u$atera tr$ange =3

(#$e)2.4

4. a$u# !" $n'$r'e !" an e*u$atera tr$ange !" #$e a= a

23

5. a$u# !" '$r'u'$r'e !" an e*u$atera tr$ange !" #$e a= a3

6. a$u# !" $n'$r'e !" a tr$ange !" area an #e$-&er$eter s=s


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A '!' ha# t! han#, the #aer !ne $# 'ae the 6our6and!r s6ort 6andh$e the arger !ne $# 'ae minute6and!r long 6and.

2.

$. n 60 $nute#, the $nute han ga$n# 55 $nute# !n theh!ur !n the h!ur han.

$$. n ever/ h!ur, b!th the han# '!$n'$e !n'e.

$$$. he han# are $n the #ae #tra$ght $ne hen the/ are'!$n'$ent !r !&&!#$te t! ea'h !ther.

$v. When the t! han# are at r$ght ange#, the/ are 15 $nute

#&a'e# a&art.

v. When the han# are $n !&&!#$te $re't$!n#, the/ are 30$nute #&a'e# a&art.

v$. Ange tra'e b/ h!ur han $n 12 hr# = 360M

v$$. Ange tra'e b/ $nute han $n 60 $n. = 360M.

v$$$. " a at'h !r a '!' $n$'ate# 8.15, hen the '!rre't t$e $#8, $t $# #a$ t! be 15 $nute# too fast.

Nn the !ther han, $" $t $n$'ate# 7.45, hen the '!rre'tt$e $# 8, $t $# #a$ t! be 15 $nute# too slow.

0aces and ames

1. 0aces:A '!nte#t !" #&ee $n runn$ng, r$$ng, r$v$ng, #a$$ng !rr!$ng $# 'ae a ra'e.

2. 0ace &ourse:he gr!un !r &ath !n h$'h '!nte#t# are ae $#'ae a ra'e '!ur#e.

3. Starting Point:he &!$nt "r! h$'h a ra'e beg$n# $# n!n a#a #tart$ng &!$nt.


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4. ,inning Point or oal:he &!$nt #et t! b!un a ra'e $# 'ae a$nn$ng &!$nt !r a g!a.

5. ,inner:he &er#!n h! r#t rea'he# the $nn$ng &!$nt $# 'aea $nner.

6. 'ead 8eat 0ace:" a the &er#!n# '!nte#t$ng a ra'e rea'h theg!a ea't/ at the #ae t$e, the ra'e $# #a$ t! be ea heatra'e.

7. Start:Su&&!#e A an are t! '!nte#tant# $n a ra'e. " be"!rethe #tart !" the ra'e, A $# at the #tart$ng &!$nt an $# ahea !" Ab/ 12 etre#, then e #a/ that GA g$ve# , a #tart !" 12 etre#G.

! '!ver a ra'e !" 100 etre# $n th$# 'a#e, A $ have t! '!ver

100 etre# h$e $ have t! '!ver !n/ (100 - 12) = 88etre#.

n a 100 ra'e, GA 'an g$ve 12 G !r GA 'an g$ve a #tart !" 12 G!r GA beat# b/ 12 G ean# that h$e A run# 100 , run#(100 - 12) = 88 .

8. ames:GA gae !" 100, ean# that the &er#!n a!ng the'!nte#tant# h! #'!re# 100 &!$nt# r#t $# the $nnerG.

" A #'!re# 100 &!$nt# h$e #'!re# !n/ 80 &!$nt#, then e #a/that GA 'an g$ve 20 &!$nt#G.

0esults on Population:

:et the &!&uat$!n !" a t!n be n! an #u&&!#e $t $n'rea#e# atthe rate !" &er annu, then

1. !&uat$!n a"ter n/ear# = 1 + n

100

2. !&uat$!n n/ear# ag! =

1 + n

100

0esults on 'epreciation:


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:et the &re#ent vaue !" a a'h$ne be . Su&&!#e $t e&re'$ate#at the rate !" &er annu. hen

1.


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tan AB

2. Trigonometrical Identities:

$. sin ! cos = 1.

$$. 1 ! tan = sec .

$$$. 1 ! cot = cosec .

3. Values of T-ratios:

0

( /6)

30

( /4)

45

( /3)

60

( /2)

90

sin 01

"

1

cos 1"

1

0

tan 01

"1 " not defined

4. Angle of Elevation:

#uppose a $an %ro$ a point O loo&s up at an o'(ect P, placed a'ove the level o% hiseye. )hen, the angle which the line o% sight $a&es with the hori*ontal through O, is

called the anlge of elevationo% P as seen %ro$ O.

Angle o% elevation o% P %ro$ O = AOP.

5. Angle of Depression:


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#uppose a $an %ro$ a point O loo&s down at an o'(ect P, placed 'elow the level o%

his eye, then the angle which the line o% sight $a&es with the hori*ontal through O,

is called the angle of depressiono% P as seen %ro$ O.

Logarit6m

1. Logarit6m:

" a$# a &!#$t$ve rea nuber, !ther than 1 an am=x, then er$tem logaxan e #a/ that the vaue !" !gxt! the ba#e a$# m.

3amples:

($). 1031000 !g101000 = 3.

($$). 34= 81 !g381 = 4.

($$$). 2-3=1

!g21

= -3.8 8

($v). (.1)2= .01 !g(.1).01 = 2.

2. Properties of Logarit6ms:

1. !ga(xy) = !gax+ !gay

2. !gax

= !gax- !gayy

3. !gx= 1


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4. !ga1 = 0

5. !ga(xn) = n(!gax)

6. !gax=

1

!ga

7. !gax=!gbx

=!gx

.!gba !g a

3. &ommon Logarit6ms:

:!gar$th# t! the ba#e 10 are n!n a# '!!n !gar$th#.

4. he !gar$th !" a nuber '!nta$n# t! &art#, nae/G'hara'ter$#t$'G an Gant$##aG.

&6aracteristic:he $nterna &art !" the !gar$th !" a nuber $#'ae $t#c6aracteristic+

a#e When the nuber $# greater than 1.

n th$# 'a#e, the 'hara'ter$#t$' $# !ne e## than the nuber !"$g$t# $n the e"t !" the e'$a &!$nt $n the g$ven nuber.

a#e When the nuber $# e## than 1.

n th$# 'a#e, the 'hara'ter$#t$' $# !ne !re than the nuber !"Ier!# beteen the e'$a &!$nt an the r#t #$gn$'ant $g$t !"the nuber an $t $# negat$ve.

n#tea !" -1, -2 et'. e r$te 1 (!ne bar), 2 (t! bar), et'.

;a&e#-

%uber hara'ter$#t$' %uber hara'ter$#t$'

654.24 2 0.6453 1

26.649 1 0.06134 2

8.3547 0 0.00123 3

.antissa:


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he e'$a &art !" the !gar$th !" a nuber $# n!n $#$t# mantissa+!r ant$##a, e !! thr!ugh !g tabe.

erutat$!n an !b$nat$!n

1. Factorial 1otation:

:et nbe a &!#$t$ve $nteger. hen, "a't!r$a n, en!te nO $# enea#

nC n(n % !)(n % 4) +++ ;+4+!+

3amples:

$. We ene "C !.

$$. 4O = (4 3 2 1) = 24.

$$$. 5O = (5 4 3 2 1) = 120.

2. Permutations:

he $>erent arrangeent# !" a g$ven nuber !" th$ng# b/ ta$ng

#!e !r a at a t$e, are 'ae &erutat$!n#.

3amples:

$. A &erutat$!n# (!r arrangeent#) ae $th theetter# a, b, cb/ ta$ng t! at a t$e are(ab ba ac ca bc cb).

$$. A &erutat$!n# ae $th the etter# a, b, cta$ng a at at$e are

(abc

acb

bac

bca

cab

cba

)

3. 1umber of Permutations:%uber !" a &erutat$!n#!" nth$ng#, taen rat a t$e, $# g$ven b/

nr= n(n- 1)(n- 2) ... (n- r+ 1) = nO

(n- r)O


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3amples:

$. 62= (6 5) = 30.

$$. 73= (7 6 5) = 210.

$$$. &or+ number of all permutations of nt6ings ta=en allat a time nC+

4. An Important 0esult:

" there are n#ube't# !" h$'hp1are a$e !" !ne $n?p2area$e !" an!ther $n?p3are a$e !" th$r $n an #! !nanprare a$e !" r

th$n,#u'h that (p1+p2+ ...pr) = n.

hen, nuber !" &erutat$!n# !" the#e n!be't# $# = nO

(p1O).(p2)O.....(prO)

5. &ombinations:

;a'h !" the $>erent gr!u !r #ee't$!n# h$'h 'an be "!re b/ta$ng #!e !r a !" a nuber !" !be't# $# 'aea combination.

3amples:

1. Su&&!#e e ant t! #ee't t! !ut !" three b!/# A, , .hen, &!##$be #ee't$!n# are A, an A.

%!te A an A re&re#ent the #ae #ee't$!n.

2. A the '!b$nat$!n# "!re b/ a, b, cta$ng ab bc ca.

3. he !n/ '!b$nat$!n that 'an be "!re !" threeetter# a, b, ctaen a at a t$e $# abc.

4.


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2 1umber of &ombinations:

he nuber !" a '!b$nat$!n# !" nth$ng#, taen rat a t$e $#

n

r=

nO

=

n(n- 1)(n- 2) ... t! r"a't!r#

.(rO)(n- rO) rO

1ote:

$. nn= 1 ann0= 1.

$$. nr=n(n - r)

3amples:

$. 114= (11 10 9 8) = 330.(4 3 2 1)

$$. 1613=16(16 - 13)=

163=16 15 14

=16 15 14

= 560.3O 3 2 1

Arit6metic Progression@en$t$!n " ea'h ter !" a #er$e# $>er# "r! $t# &re'e$ng ter b/ a'!n#tant, then #u'h a &r!gre##$!n $# 'ae ar$thet$' &r!gre##$!n.

e+g: 4DE!!n the ab!ve #e*uen'e the $>eren'e beteen an/ t! ter# $n 3.!rua#!n#$er an A a (a # d) (a # 4d) (a # ;d)(a # 5d)++++1) n th ter !" the ab!ve #e*uen'e = *n a # (n % !)d2) Su t! n n! !" ter# Sn n / 4 (4a # (n % !)d)3) An!ther "!rua "!r Sn n / 4 (rst term # last term)

eometric Progression+@en$t$!n An/ &r!gre##$!n !r #e*uen'e $th #u''e##$ve ter# hav$ng a'!!n rat$!n $# 'ae geometric progression+e+g: 45E!+++

n the ab!ve #e*uen'e the '!!n rat$! $# 2 "!r an/ t! #u''e##$ve ter#!rua#!n#$er a - a ar ar4++++ here a $# the r#t ter an r $# the '!!nrat$!4) n th ter, *n arn%!

5) Su t! n ter# , Sn a(! % rn)/(!%r)6) n 'a#e r $e# beeen -1 an +1 then Sum of all terms of t6e series a/(! % r)


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8&F and L&.r!u't !" t! nuber# = he$r B E he$r :J

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Aapti by Amar