Chapter 13.9, Problem 2GP

To determine

The number of $7$ -digit identification number is formed with $1,2,3,5,6,8,9$ if no number is repeated is to be estimated.

Answer to Problem 2GP

From $1,2,3,5,6,8,9$ total $5040$ numbers are formed which are $7$ -digit identification number.

Explanation of Solution

Given:

  $\begin{array}{l}n=7\\ r=\left\{1,2,3,5,6,8,9\right\}=7\end{array}$

Concept Used:

The formula for combination is given by the following expression:

  $C{}_r=\frac{n!}{\left(n-r\right)!}$

Calculation:

The number of $7$ -digit identification number is calculated as:

  $C{}_r=\frac{n!}{\left(n-r\right)!}$

Put the given data in the above formula:

  $\begin{array}{l}C{}_7=\frac{7!}{\left(7-7\right)!}\\ C{}_7=\frac{7!}{0!}\\ C{}_7=\frac{7\times 6\times 5\times 4\times 3\times 2\times 1}{1}\left[\because 0!=1\right]\\ C{}_7=5040\end{array}$

Conclusion:

Hence, the count of $7$ -digit identification number are $C{}_4=5040$