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The music class consists of 4 girls and 7 boys. How many way

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The music class consists of 4 girls and 7 boys. How many way [#permalink] New post 16 Oct 2007, 11:50
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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
[Reveal] Spoiler: OA

Last edited by Bunuel on 30 Jun 2012, 04:04, edited 1 time in total.
Edited the question.
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Re: PS Combinatorics - Music Class [#permalink] New post 16 Oct 2007, 12:02
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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 [#permalink] New post 16 Oct 2007, 12:11
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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Re: [#permalink] New post 29 Jun 2012, 13:47
jimjohn wrote:
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


I don't really understand how Case 2 and 3 above are calculated. Any help please?


Thanks,
Diana
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Re: Re: [#permalink] New post 30 Jun 2012, 04:03
dianamao wrote:
jimjohn wrote:
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


I don't really understand how Case 2 and 3 above are calculated. Any help please?


Thanks,
Diana


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

Reverse Approach:

The # of groups with at least one boy equal to total groups of 3 that can be formed out of 11 people minus groups with all girls: C^3_{11}-C^3_4=161.

Answer: C.

Direct Approach:

The # of groups with at least one boy equal to groups with one boy (and 2 girls) plus groups with 2 boys (and 1 girl) plus 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: The music class consists of 4 girls and 7 boys. How many way [#permalink] New post 30 Jun 2012, 04:50
Awesome. Thanks Bunuel!

Cheers,
Diana
Re: The music class consists of 4 girls and 7 boys. How many way   [#permalink] 30 Jun 2012, 04:50
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