Question
How many $5$ digit numbers can be formed using all the digits $3, 8, 0, 7, 6 ?$
✓
Answer
To form $5$ digit numbers using all the digits $3,8.0 .7,6$.
$0$ should not be in the first place.
$\therefore$ From the remaining $4$ digits $(3,8,7,6)$ one digit can be placed at the first placed in ${ }^4 P_1$ ways.
Now, remaining $4$ digits including $0$ , can be arranged in remaining $4$ places in ${ }^4 P_4$ ways.
First place Remaining $4$ places
$\therefore$ Total permutations $={ }^4 P_1 \times{ }^4 P_4=4 \times 4!=4 \times 24=96$
$0$ should not be in the first place.
$\therefore$ From the remaining $4$ digits $(3,8,7,6)$ one digit can be placed at the first placed in ${ }^4 P_1$ ways.
Now, remaining $4$ digits including $0$ , can be arranged in remaining $4$ places in ${ }^4 P_4$ ways.
First place Remaining $4$ places
$\therefore$ Total permutations $={ }^4 P_1 \times{ }^4 P_4=4 \times 4!=4 \times 24=96$
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