Twelve jurors must be picked from a pool of n potential

The answer is D.

Essentially, there is an (n-m) pool of jurors to select a group of 12 from. As a result, the number of possible juries is equivalent to \(_{(n-m)}C_{12}\), or:

\(\frac{(n-m)!}{12!*(n-m-12)!}\)

For simplicity's sake, let's set X = n - m (the pool of jurors available after processing by the defense counsel). As a result the number of possible juries is equivalent to:

\(\frac{X!}{12!*(X-12)!}\)


Statement 1: If one less potential juror had been rejected ,it would be possible to create 13 different juries:

As a result, X increases to X + 1:

\(\frac{(X+1)!}{12!*(X+1-12)!} = 13\)

\(\frac{(X+1)!}{12!*(X-11)!} = 13\)

\(\frac{(X+1)!}{(X-11)!} = 13*12!\)

\(\frac{(X+1)!}{(X-11)!} = 13!\)

\((X+1)*(X)*(X-1)*...*(X-10) = 13!\)

Therefore, by using our understanding of factorials, we can determine X (or n-m) to be 12. The answer to the original question then becomes 1, and Statement 1 is proven to be sufficient.

Statement 2: n=m+12:

Rearranging the equation sets n-m = 12, which is enough to show that the answer to the original question is 1 possible jury, and Statement 2 also proves to be sufficient.

Therefore, both statements are sufficieint by themselves, and the answer is D.