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Q. The number of all numbers having $5$ digits, with distinct digits is

WBJEEWBJEE 2017Permutations and Combinations

Solution:

Total number of $5$ -digit numbers having all the digits distinct $={ }^{10} P _{5}-{ }^{9} P_{4}$
$=\frac{10 !}{5 !}-\frac{9 !}{5 !}=\frac{10 \times 9 !}{5 !}-\frac{9 !}{5 !} $
$=\frac{9 !}{5 !}(10-1)=\frac{9 !}{(9-4) !}(9) $
$=9 \times{ }^{9} P_{4}$