Question

A $7$-digit number is formed using the digits from $1,2,3,\cdots ,9$ such that exactly one digit is repeated thrice and exactly one digit is repeated twice. The total number of such $7$-digit number is

• A. $9\times 9\times 9\times 8\times 8\times 7\times 6$
• B. $\frac{{}^9{C}_1\times {}^8{C}_1\times {}^7{C}_5\times 7!}{3!\times 2!}$
  
• C. $\frac{{}^9{C}_2\times {}^7{C}_5\times 7!}{5!}$
• D. $\frac{7!}{3!\times 2!}$

Answer 1

The correct option is B $\frac{{}^9{C}_1\times {}^8{C}_1\times {}^7{C}_5\times 7!}{3!\times 2!}$


Number of ways of selecting one digit which is repeated thrice is ${}^9{C}_1$
Number of ways of selecting one digit which is repeated twice is ${}^8{C}_1$
Number of ways of selecting remaining digits is ${}^7{C}_2$

Total possible numbers $=\frac{{}^9{C}_1\times {}^8{C}_1\times {}^7{C}_2\times 7!}{3!\times 2!}$

(Divided by $3!\times 2!$ because one digit is repeated $3$ times and one digit is repeated $2$ times)