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
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