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If a music class consists of 4 girls and 7 boys, how many ways can

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bmwhype2
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If a music class consists of 4 girls and 7 boys, how many ways can [#permalink] New post   Updated on: 17 Oct 2023, 02:17
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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

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C

Originally posted by bmwhype2 on 16 Oct 2007, 11:50.
Last edited by Bunuel on 17 Oct 2023, 02:17, edited 2 times in total.
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bkk145
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Re: If a music class consists of 4 girls and 7 boys, how many ways can [#permalink] New post   16 Oct 2007, 12:02
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bmwhype2 wrote:
The music class consists of 4 girls and 7 boys. How many ways can a group of 3 be formed if it has to include at least one boy?

155
158
161
165
172


Classic combination problem
At least 1 boy = Total - all girls
All girls = C(4,3) = 4
Total combination = C(11,3) = 165
Ans = 165-4 = 161
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Re: If a music class consists of 4 girls and 7 boys, how many ways can [#permalink] New post   16 Oct 2007, 12:11
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Case 1: All three boys
(7 choose 3) = 35

Case 2: Two boys, one girl
(7 choose 2) * (4 choose 1) = 84

Case 3: One boy, two girls
(7 choose 1) * (4 choose 2) = 42


Add them up 35 + 84 + 42 = 161
C
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If a music class consists of 4 girls and 7 boys, how many ways can [#permalink] New post   30 Jun 2012, 04:03
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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


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
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Re: If a music class consists of 4 girls and 7 boys, how many ways can [#permalink] New post   30 Jun 2012, 04:50
Awesome. Thanks Bunuel!

Cheers,
Diana
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Re: If a music class consists of 4 girls and 7 boys, how many ways can [#permalink] New post   26 Sep 2017, 16:25
Expert Reply
bmwhype2 wrote:
The music class consists of 4 girls and 7 boys. How many ways can a group of 3 be formed if it has to include at least one boy?

A. 155
B. 158
C. 161
D. 165
E. 172



We can determine the number of ways a group of 3 that includes at least 1 boy can be formed from 4 girls and 7 boys using the following formula:

Total number of ways to form a group of 3 - number of ways to form the group with no boys

Total number of ways to form a group of 3 is 11C3 = 11!/[3!(11-3)!] = (11 x 10 x 9)/3! = (11 x 10 x 9)/(3 x 2 x 1) = 11 x 5 x 3 = 165

Number of ways to form the group with no boys = 4C3 = 4!/[3!(4-3)!] = (4 x 3 x 2)/3! = 4

Thus, the total number of ways to select the group with at least 1 boy is 165 - 4 = 161.

Answer: C
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Re: If a music class consists of 4 girls and 7 boys, how many ways can [#permalink] New post   09 Dec 2023, 20:34
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Re: If a music class consists of 4 girls and 7 boys, how many ways can [#permalink]
09 Dec 2023, 20:34
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