Question

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

Answer 1

Hint: The numbers of jurors can be calculated by using the combination formula from the formulae for permutation and combination. The combination helps us in selecting a given number of objects out of a total number of given objects but does not account for the arrangement of those objects in the selection. Since when selecting jurors the order of their selection is certainly not important we will use a combination formula instead of a permutation formula. The combination formula is as follows:
For combination of $ r $ given objects out of $ n $ given objects formula for combination is given as:
 $ {}^n{c_r} = \dfrac{{n!}}{{r!(n - r)!}} $ do not hear that $ ! $ here stands for factorial.

Complete step-by-step answer:
Since we have to find the combination of $ r $ objects out of $ n $ given objects we will use the combination formula which is given by
 $ {}^n{c_r} = \dfrac{{n!}}{{r!(n - r)!}} $
We see here that the $ n $ given here is $ 50 $ and value of $ r $ given here is $ 12 $ so applying the formula we get
 $ {}^{50}{c_{12}} = \dfrac{{50!}}{{12!(38)!}} $
Upon solving we get
 $ {}^{50}{c_{12}} = \dfrac{{50\times49\times48\times47\times46\times45\times45\times44\times43\times42\times41\times40\times39\times38!}}{{12!(38)!}} $
Which will then be written as
\[{}^{50}{c_{12}} = \dfrac{{50\times49\times48\times47\times46\times45\times45\times44\times43\times42\times41\times40\times39}}{{12!}}\]
Upon solving the above multiplication we get:
\[121399651100\]
Thus the jurors can be selected in these numbers of ways.
So, the correct answer is “\[121399651100\]”.

Note: Do always note that we use the formula for combination of elements when there is no need for us to consider the order in which they are going to be selected while in permutation order in which the objects are selected is very important. The above question therefore uses combination instead of permutation.