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

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CEO
Joined: 21 Jan 2007
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The music class consists of 4 girls and 7 boys. How many way  [#permalink]

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Updated on: 30 Jun 2012, 04:04
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80% (01:29) correct 20% (02:18) wrong based on 366 sessions

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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

Originally posted by bmwhype2 on 16 Oct 2007, 11:50.
Last edited by Bunuel on 30 Jun 2012, 04:04, edited 1 time in total.
Edited the question.
VP
Joined: 10 Jun 2007
Posts: 1391
Re: PS Combinatorics - Music Class  [#permalink]

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16 Oct 2007, 12:02
1
1
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
Manager
Joined: 19 Aug 2007
Posts: 164

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16 Oct 2007, 12:11
2
1
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
Intern
Joined: 25 Jun 2011
Posts: 37
Location: Sydney

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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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Joined: 02 Sep 2009
Posts: 49251

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30 Jun 2012, 04:03
1
1
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: $$C^3_{11}-C^3_4=161$$.

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: $$C^1_{7}*C^2_4+C^2_{7}*C^1_4+C^3_{7}=161$$.

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Re: The music class consists of 4 girls and 7 boys. How many way  [#permalink]

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30 Jun 2012, 04:50
Awesome. Thanks Bunuel!

Cheers,
Diana
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Joined: 06 Sep 2013
Posts: 1821
Concentration: Finance
Re: The music class consists of 4 girls and 7 boys. How many way  [#permalink]

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30 Dec 2013, 07:34
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

Reverse combinatorics approach is preferred method on this one
Note that at least 1 boy gives clue to using this method

All combinations - four girls = answer

All combinations is 11C3 = 165

All girls 4C3 = 4

So our answer is 165-4 = 161

Hope it helps
Cheers!

J
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Joined: 06 Nov 2014
Posts: 1887
Re: The music class consists of 4 girls and 7 boys. How many way  [#permalink]

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31 Mar 2015, 12:47
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 want At least 1 boy = Total - all girls
Now all 3 girls can be selected in 4C3 = 4 ways
No. of ways in which 3 people can be selected out of 11 = !!C3 = 165
Required number of ways = 165 - 4
= 161.

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Re: The music class consists of 4 girls and 7 boys. How many way  [#permalink]

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31 Mar 2015, 13:13
I often confuse distinct between n!/((n-k)!*k!) and simple n!/(n-k)!. In this case I spend 90 seconds using solving 3 components, got answer 420, find out that it s wrong and divided every component to k!. And I did it with a simple example (how many variants to take 2 from 4) That is not good, I suppose) Too much time, and big risk to make mistake. What can you advice?
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Re: The music class consists of 4 girls and 7 boys. How many way  [#permalink]

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25 Sep 2017, 05:19
music class
4 girls
7 boys
total = 11
3 is the number of group to be formed

using this formula: n!(n-r!)r! therefore 11! - 7!
- (11-3!)3! (7-3!)3!

= 165 - 4 = 161
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Re: The music class consists of 4 girls and 7 boys. How many way  [#permalink]

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26 Sep 2017, 16:25
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.

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Re: The music class consists of 4 girls and 7 boys. How many way &nbs [#permalink] 26 Sep 2017, 16:25
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