Problem Solving

Quant 2 Q 160

A couple decides to have 4 children. If they succeed in having 4 children and each child is equally likely to be a boy or a girl, what is the probablity that they will have exactly 2 girls and 2 boys?

A) 3/8
B) 1/4
C) 3/16
D) 1/8
E) 1/16

I understand the answer should be number of possibilities of 2 girls 2 boys, which is 4!/2!2!, divided by total possibilities. Rather than list these out, how can we use the combination formula to figure out total number of possibilities (simply, its 2x2x2x2, but i'm trying to figure out if a question is way more complex than this what is formula for total number of combinations possible)

zank wrote: I understand the answer should be number of possibilities of 2 girls 2 boys, which is 4!/2!2!, divided by total possibilities. Rather than list these out, how can we use the combination formula to figure out total number of possibilities (simply, its 2x2x2x2, but i'm trying to figure out if a question is way more complex than this what is formula for total number of combinations possible)

No of possibilities for 1st child = 2 [boy/girl]
similarly, 2nd, 3rd and 4th will each have 2 possibilites
Making the total = 2*2*2*2

zank wrote:Rather than list these out, how can we use the combination formula to figure out total number of possibilities (simply, its 2x2x2x2, but i'm trying to figure out if a question is way more complex than this what is formula for total number of combinations possible)

The formula is 2^n where n is the number of children born. Obviously this can apply to other situations too where the number of possible outcomes for each event is exactly two, such as the number of outcomes for n coin tosses. Calculating the number of total outcomes isn't a combinations problem, so there's no simple, straightforward application of the combinations formula that will yield this number.

In any case, you're probably better off understanding the fundamental logic behind this formula rather than memorizing it, as most difficult GMAT counting problems resist a purely formulaic approach.

Each child has 2 options of being a boy or a girl.
Thus the total # of outcomes will be 2 ^ 4 by multiplication rule of counting.