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Permutations — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsPermutations2 Marks
Question
How many three-digit numbers are there, with no digit repeated?
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Answer

Total number of digits = 10
Total number of 3 digit numbers $= \ ^{10}\text{P}_3 $
But these arrangements also indude those numbers which have O at hundred's place. such numbers are not 3-digit numbers.
When 0 is fixed at hundred's place, we have to arrange remaining 9 digits by taking 2 at a time.
The number of such arrangements is $= \ ^9\text{P}_3$
So, the total of numbers having O at hundred's place $= \ ^9\text{P}_2$
Hence, total number of 3 digit numbers which distinct $= \ ^{10}\text{P}_3 - ^{9}\text{P}_2 $
$=\frac{10!}{(10-3)!}-\frac{9!}{(9-2)!}$
$=\frac{10!}{7!}-\frac{9!}{7!}$
$=\frac{10\times9\times8\times7!}{7!}-\frac{9\times8\times7!}{7!}$
$= 720-72$
$=648$

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