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16 Sep 2014, 01:08
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If a music class consists of 4 girls and 7 boys, how many ways can a group of 3 be formed if it must include at least one boy?
A. 155
B. 158
C. 161
D. 165
E. 172
A. 155
B. 158
C. 161
D. 165
E. 172
16 Sep 2014, 01:08
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Official Solution:
If a music class consists of 4 girls and 7 boys, how many ways can a group of 3 be formed if it must include at least one boy?
A. 155
B. 158
C. 161
D. 165
E. 172
REVERSE APPROACH:
The number of groups with at least one boy is equal to the total number of groups of 3 that can be formed from 11 students minus the groups consisting solely of girls:
\(C^3_{11}-C^3_4=161\).
DIRECT APPROACH:
The number of groups with at least one boy is the sum of the groups with one boy (and 2 girls), the groups with 2 boys (and 1 girl), and the groups with 3 boys:
\(C^1_{7}*C^2_4+C^2_{7}*C^1_4+C^3_{7}=161\).
Answer: C
If a music class consists of 4 girls and 7 boys, how many ways can a group of 3 be formed if it must include at least one boy?
A. 155
B. 158
C. 161
D. 165
E. 172
REVERSE APPROACH:
The number of groups with at least one boy is equal to the total number of groups of 3 that can be formed from 11 students minus the groups consisting solely of girls:
\(C^3_{11}-C^3_4=161\).
DIRECT APPROACH:
The number of groups with at least one boy is the sum of the groups with one boy (and 2 girls), the groups with 2 boys (and 1 girl), and the groups with 3 boys:
\(C^1_{7}*C^2_4+C^2_{7}*C^1_4+C^3_{7}=161\).
Answer: C
17 Oct 2023, 02:15
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
