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(A) $5$ digit telephone number starts with $67$. Since the first two digits are fixed as $6$ and $7$,we need to fill the remaining $3$ positions. The

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$336$

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(A) $5$ digit telephone number starts with $67$. Since the first two digits are fixed as $6$ and $7$,we need to fill the remaining $3$ positions. The available digits are ${0, 1, 2, 3, 4, 5, 8, 9}$,which are $8$ distinct digits. Since no digit can be repeated,we need to arrange $3$ digits out of these $8$ available digits. The number of ways to arrange $3$ digits out of $8$ is given by the permutation formula $^nP_r = \frac{n!}{(n-r)!}$. Here,$n = 8$ and $r = 3$. $^8P_3 = \frac{8!}{(8-3)!} = \frac{8!}{5!} = 8 	imes 7 	imes 6 = 336$.

How many $5$ digit telephone numbers can be constructed using the digits $0$ to $9$,if each number starts with $67$ and no digit appears more than once?

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