Grass Seed Problem A landscaper wanted to sow a blend of grass seed on a homeowner’s lawn. The...

Grass Seed Problem

A landscaper wanted to sow a blend of grass seed on a homeowner’s lawn. The landscaper had available two boxes each of 4 different types of grass seed. If the landscaper could only mix full boxes of the grass seeds in the blend, how many different blends of grass seed could the landscaper use?

Consider

A B C D

A B C D

AA = A

AABB = AB

AABBCCDD = ABCD

Some note:

AABBCC = ABC

If ONE bag is selected

1000

!3!1

!4

4total

4

!2!2

!4

If TWO bags are selected

1100 2000

6total

This is the same as

1000

6

1110

If THREE bags are selected

2100

16total

!1!3

!4

!2!1!1

!4

4 12

2200

If FOUR bags are selected

1111 2110

13total

This is the same as

1100

!4

!4

!1!2!1

!4

1 12

If FIVE bags are selected

2111 2210

16total

!3!1

!4

!1!1!2

!4

4 12

If SIX bags are selected

2211

6total

!2!2

!4

6

2220

This is the same as

1110

If SEVEN bags are selected

2221

4total

!1!3

!4

4

If EIGHT bags are selected

2222

0total

This is the same as

1111

Summing it up……1 1000 = 42 1100 2000 = 63 2100 1110 = 12 + 4 = 164 2111 1111 2200 = 12 + 1 = 135 2210 2111 = 12 + 4 = 166 2211 2220 = 67 2221 = 48 2222 = 0

--------

TOTAL 65

Grass Seed Problem

A landscaper wanted to sow a blend of grass seed on a homeowner’s lawn. The landscaper had available two boxes each of 4 different types of grass seed. If the landscaper could only mix full boxes of the grass seeds in the blend, how many different blends of grass seed could the landscaper use?

If ONE bag is selected

1000

414 C

4total

If TWO bags are selected

1100

624 C

2000

6total

This is the same as

1000

1110

If THREE bags are selected

434 C

2100

12261224 CC

16total

2200

If FOUR bags are selected

1111

144 C

2110

12341334 CC

13total

This is the same as

1100

If FIVE bags are selected

2111

41444 CC

2210

12342334 CC

16total

If SIX bags are selected

2211

6612444 CC

2220

6total

This is the same as

1110

If SEVEN bags are selected

2221

4413444 CC

6total

If EIGHT bags are selected

2222

0total

This is the same as

1111

Summing it up……1 1000 C(4, 1) = 42 1100 C(4, 2) = 63 2100 1110 C(4, 2) C(2, 1) + C(4, 3)

= 6 * 2 + 4 = 164 2111 1111 C(4, 3) C(3, 1) + C(4, 4)

= 4 * 3 + 1 = 135 2210 2111 C(4, 3) C(3, 2) + C(4, 4) C(4, 1)

= 4 * 3 + 1* 4 = 166 2211 C(4, 4) C(4, 2)

= 1 * 6 = 67 2221 C(4,4) C(4,3)

= 1 * 4 = 48 = 0

--------

TOTAL 65

Grass Seed Problem

A landscaper wanted to sow a blend of grass seed on a homeowner’s lawn. The landscaper had available three boxes each of 4 different types of grass seed. If the landscaper could only mix full boxes of the grass seeds in the blend, how many different blends of grass seed could the landscaper use?

Consider

A B C D

A B C D

A B C D

AA = A

AAABBB = AB

AAABBBCCCDDD = ABCD

Some note:

If ONE bag is selected

1000

!3!1

!4

4total

4

!2!2

!4

If TWO bags are selected

1100 2000

6total

This is the same as

1000

4

2200

If FOUR bags are selected

1111 2110

25total

This is the same as

1100

!4

!4

!1!2!1

!4

1 12

3100

!2!1!1

!4

12

1110

If THREE bags are selected

2100

16total

3000

This is the same as

1000

!1!3

!4

!2!1!1

!4

4 12

3200

If FIVE bags are selected

2111 2210

40total

!3!1

!4

!1!1!2

!4

3110

!1!2!1

!4

4 12 12

!2!1!1

!4

12

3300

If SIX bags are selected

3111 2211

34total

!3!1

!4

!2!2

!4

3210

!1!1!1!1

!4

4 6 24

This is the same as

1100

2221

If SEVEN bags are selected

3310 3220

40total

!1!1!2

!4

!1!2!1

!4

3211

!2!1!1

!4

12 12 12

!1!3

!4

4

2222

If EIGHT bags are selected

3320 3311

30total

!1!1!2

!4

!2!2

!4

3221

!1!2!1

!4

12 6 12

This is the same as

1111

If NINE bags are selected

3330 3321

16total

!1!1!2

!4

3222

!3!1

!4

12 4

This is the same as

1110

If TEN bags are selected

3331 3322

10total

!1!3

!4

!2!2

!4

4 6

If ELEVEN bags are selected

3332

4total

!1!3

!4

4

If TWELVE bags are selected

3333

0total

This is the same as

1111

Summing it up…… 1 1000 = 4 2 1100 = 6 3 2100 1110 = 16 4 3100 2110 1111 = 25 5 3200 3110 2210 2111 = 40 6 3210 3111 2211 = 34 7 3310 3220 3211 2221 = 40 8 3320 3311 3221 = 30 9 3321 3222 = 1610 3331 3322 = 1011 3332 = 412 = 0

--------

TOTAL 225

Grass Seed Problem

A landscaper wanted to sow a blend of grass seed on a homeowner’s lawn. The landscaper had available two boxes each of 5 different types of grass seed. If the landscaper could only mix full boxes of the grass seeds in the blend, how many different blends of grass seed could the landscaper use?

10000

If ONE bag is selected

!4!1

!5

5total

5

11000

!3!2

!5

If TWO bags are selected

20000

10total

This is the same as 10000

10

2100011100

If THREE bags are selected

30total

!2!3

!5

!3!1!1

!5

10 20

2110011110 22000

If FOUR bags are selected

35total

This is the same as 11000

!1!4

!5

!2!2!1

!5

5 30

2111011111

If FIVE bags are selected

51total

!5

!5

!1!3!1

!5

1 20

22100

!2!1!2

!5

30

2211021111 22200

If SIX bags are selected

35total

This is the same as 11100

!4!1

!5

!1!2!2

!5

5 30

2221022111

If SEVEN bags are selected

30total

!3!2

!5

!1!1!3

!5

10 20

22211

!2!3

!5

If Eight bags are selected

22220

10total

This is the same as 11110

10

22221

If NINE bags are selected

!1!4

!5

5total

5

If TEN bags are selected

22222

0total

This is the same as 11111

Summing it up……1 10000 = 5 = 52 11000 = 10 = 103 11100 21000 = 10 + 20 = 304 11110 21100 = 5 + 30 = 355 11111 21110 22100 = 1 + 20 + 30 = 516 21111 22110 = 5 + 30 = 357 22111 22210 = 10 + 20 = 308 22211 = 10 = 109 22221 = 5 = 510 = 0

--------

TOTAL 211