
Aniket D. answered 02/02/24
T15 Medical Student | 100th Percentile MCAT 1st Attempt
Let there be n total mice in the litter. We know that there are 2 white mice, and that means that there are (n-2) other mice in the litter.
In this solution, P(X) means "the probability of X occurring.
In this solution, nCx means n choose x (the number of combinations of choosing x items from n total items)
P(both white mice chosen) = (choosing the 2 white mice and choosing 2 more random mice from the remaining litter)/(total number ways to choose 4 mice)
Once the 2 white mice are chosen, there are n-2 mice remaining, of which 2 will be randomly chosen:
P(both white mice chosen) = (n-2)C2/(nC4) ... (A)
P(neither white mouse chosen) = (choosing all 4 mice from the non-white mice portion of the litter)/(total number of ways to choose 4 mice)
P(neither white mouse chosen) = (n-2)C4/(nC4) ...(B)
We know that equation (A) = 2* equation (B)
(n-2)C2/(nC4) = 2* (n-2)C4/(nC4)
The denominators of both sides of this equation are the same, and can be canceled out. This leaves us:
(n-2)C2 = 2* (n-2)C4
Expanding both sides as suggested by the combinations formula: nCr = n!/(r! * (n-r)!)
We can also cancel some terms and clean up both sides:
(n-2)(n-3)/(2*1) = 2* (n-2)(n-3)(n-4)(n-5)/(4*3*2*1)
dividing both sides by (n-2)(n-3) and multiplying by 4! gives us:
12 = 2 * (n-4)(n-5)
dividing both sides by 2 and expanding the right hand side gives us:
6 = n2 - 9n + 20
solving this quadratic equation gives us:
(n-2)(n-7) = 0
n = 2, 7
We know that a total of 4 mice are chosen from the litter, so n (the total number of mice) cannot be less than 4. Therefore, the only possible answer is 7 total mice in the litter.