Question

10) A jury pool of consists of 50 potential jurors. In how many ways can a jury of 12 be selected?

10) A jury pool of consists of 50 potential jurors. In how many ways can a jury of 12 be selected?

Answer 1

`""_50C_12` ways.

`(""_50C_12 = 121,399,651,100)`

Explanation:

We have:

50 choices for the first member,
49 choices for the second member,
....etc.
40 choices for the 11th member, and
39 choices for the 12th member.

If we want to keep the jurors in the order we picked them, we have

`color(white)= 50xx49xx48xx...xx40xx39`

`=(50!)/(38!)`

`=(50!)/((50-12)!)`

ways to create an ordered jury.

This can be written as `""_50P_12` to indicate we are permuting 12 units from a population of 50 (order matters).

But, since order does not matter in this case (i.e. all jurors have the same "rank"), we need to divide this by the number of ways these 12 jurors can be ordered. That number is `12!`.

The number of juries possible is then:

`color(white)= (50!)/(38!)-:12!`

`=(50!)/((50-12)!xx12!)`

This can be written as `""_50C_12` to indicate we are combining 12 units from a population of 50 (order does not matter).

Answer 2

`121,399,651,100`

Explanation:

Classic combinatorics problem. To enumerate the number of ways `k` items can be chosen from `n` total items we calculate

`(n!)/(k!(n-k)!)=((50!)/(12!*38!))`

`=(50*49*48*47*46*45*44*43*42*41*40*39)/(12*11*10*9*8*7*6*5*4*3*2)`

Prime factorization and cancelling show that this is equal to

`47*43*41*23*13*7^2*5^2*2^2=121,399,651,100`