In a group of 6 boys and 4 girls, four children are to be selected.

Question

In a group of 6 boys and 4 girls, four children are to be selected. In how many different ways can they be selected such that at least one boy should be there?

A. 159
B. 208
C. 209
D. 212
E. 215

Bunuel · Accepted Answer

(At least 1 boy) = (Total) - (0 boys, so all girls)

(At least 1 boy) = 10C4 - 4C4 = 10!/(6!4!) - 1 = 209.

Answer: C.

gmatzpractice · Answer

10 C 4
1 way to choose no boys 
10 C 4 - 1 
Answer C. 209

ScottTargetTestPrep · Answer

We can use the equation:

Total number of ways to select a group of 4 children - number of ways to select a group with no boys = number of was to select a group with at least 1 boy

Total number of ways to select a group of 4 children is 10C4:

10C4 = 10!/(4! x 6!) = (10 x 9 x 8 x 7)/(4 x 3 x 2 x 1) = 10 x 3 x 7 = 210

Number of ways to select a group with no boys (i.e., all girls) is 6C0 x 4C4 = 1 x 1 = 1.

Thus, the number of was to select a group with at least 1 boy is 210 - 1 = 209.

Answer: C

aaa2007 · Answer

Thank you for the explanation Bunuel!

However, could you please explain what is wrong in the below approach: atleast one guy - 6C1 multiplied by 9C3 - ways to select three people from the remaining 9. The above approach gives a wrong answer but i am unable to identify the error in logic. Please help!

Bunuel · Answer

This way will give you duplication.

Say, the boys are {1, 2, 3, 4, 5, 6} and girls are {a, b, c, d}.

6C1 gives you one boy from six, say {1}.
9C3 gives you three children from nine, say {2, 3, a}.
So, the group is {1, 2, 3, a}.

But we will get the same group in other ways too. For example:
6C1 can give you one boy from six, say {2}.
9C3 can give you three children from nine, say {1, 3, a}.
So, the group is {1, 2, 3, a}.

As you can see we got the same group of children.

Seryozha · Answer

One can easily grasp these kinds of questions simply by imagining the concept on her/his own. What does it mean to have at least 1 boy? It means that the problem talks about 4 possible scenarios: 1 boy 3 girls, 2 boys 2 girls, 3 boys 1 girl, and 4 boys and no girl. 
So,
1st scenario: 6C1 * 4C3
2nd scenario: 6C2 * 4C2
3rd scenario: 6C3 * 4C1
4th scenario: 6C4 * 4C0
And finally, add these 4 and get the answer.

AkshdeepS · Answer

Why we used 4C4 and not 4!. Isn't 4! ways to choose 4 people from a pool of the 4 people?

Bunuel · Answer

No. 4! is the number of permutations of 4 distinct items.

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In a group of 6 boys and 4 girls, four children are to be selected.

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