Question

How many ways can a committee of 4 people be selected from a group of 7 people?

Answer 1

Hint: According to given in the question we have to determine the number of ways can a committee of 4 people be selected from a group of 7 people Step by step solution. So, first of all we have to use the permutation and combination to obtain the required number of ways for which we have to choose the required number of people from the total number of people.
Now, as mentioned in the question that there are total 7 people in which we have to make the committee of 4 people so for this selection we have to user the formula which is as mentioned below:
Formula used:
$ \Rightarrow c_r^n = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}.................(A)$
Where, n is the total number of ways or we can say that n is the total number of people in the given group and r is the required number of ways.
Now, to solve the factor of a given number n we have to use the formula to find the factor which is as mentioned below:
Formula used:
$ \Rightarrow n! = n(n - 1) \times (n - 2) \times ..... \times 2 \times 1...............(B)$
Hence, with the help of the formula (B) we can determine the value factor for the given number.

Complete step-by-step answer:
Step 1: First of all we have to use the permutation and combination to obtain the required number of ways for which we have to choose the required number of people which are 4 from the total number of people which are 7.
Step 2: Now, as mentioned in the question, there are a total 7 people in which we have to make a committee of 4 people so for this selection we have to use the formula (A) which is as mentioned in the solution hint. Hence,
$
   \Rightarrow c_4^7 = \dfrac{{7!}}{{4!\left( {7 - 4} \right)!}} \\
   \Rightarrow c_4^7 = \dfrac{{7!}}{{4! \times 3!}}............(1) \\
 $
Step 3: Now, to solve the expression (1) as obtained in the solution step 2 we have to use the formula (B) which is used to determine the factorial of a given number. Hence,
$
   \Rightarrow c_4^7 = \dfrac{{7 \times 6 \times 5 \times 4!}}{{4! \times 3!}} \\
   \Rightarrow c_4^7 = \dfrac{{7 \times 6 \times 5}}{{3 \times 2 \times 1}} \\
 $
On solving the expression as obtained just above,
$ \Rightarrow c_4^7 = 35$

Hence, with the help of the formula (A) and (B) we have determined the required number of ways a committee of 4 people can be selected from a group of 7 people is 35.

Note:
To find the factor of a given number we just have to subtract the number by 1 until the given number becomes 1. We can also understand it with the help of the formula to find the factorial which is as mentioned in the solution hint.
To choose the number of ways we always have to use the permutation and combination to obtain the required number of ways from the given total number of ways.