Question

How many different 4-persons committees can be chosen from the 100 members of the Senate?
A. 25
B. 400
C. 3,921,225
D. 94,109,400

Answer 1

Hint: Here, we can use combination rules and formulas, as we have to choose 4 different persons from 100 persons. Apply the combination formula, and simplify to get the answer.

Complete step by step answer:
Given, total number of members = 100
Number of members in committee = 4
Choose 4 different persons out of 100 is given as ${}^{100}{C_4}$
[Combination Formula: If are choosing r items out of total n items, then it can be done in ${}^n{C_r}$ways]
Here, n = 100, r = 4
So, total number of ways
= ${}^n{C_r} = {}^{100}{C_4} = \dfrac{{100!}}{{4!(100 - 4)!}} = \dfrac{{100!}}{{4!96!}}$
100! = 100 × 99 × 98 × 97 × 96 × 95 × 94 × …× 3 × 2 × 1
96! = 96 × 95 × 94 × …× 3 × 2 × 1
4! = 4 × 3 × 2 × 1
Putting values,
$\dfrac{{100!}}{{4!96!}} = \dfrac{{100 \times 99 \times 98 \times 97}}{{4 \times 3 \times 2 \times 1}} = 25 \times 33 \times 49 \times 97 = 3921225$

Total number of different 4-person committees can be chosen from the 100 members of the Senate is 3,921,225.Hence, option (C) is correct.

Note:
In these types of questions, first check whether a question is asked about combination or permutation. Permutation means arrangement of things, and combination means taking a particular number of items at a time (arrangement does not matter in combination). Then apply the proper formula as required i.e. combination or permutation. Also, if you get a factorial of a large number, never find the value of that factorial. Cancel the two factorials using proper rules. If in any case you get the final result in factorial of a large number, then leave the result in factorial form, don’t find the value.