Question

How many numbers of five digits can be made with the digits $1,2,3$ each of which can be used at most thrice in a number?

Answer 1

The five digit number can be of the format of aaabc or aabbc

where a,b,c are the digits

$Case-1\left(aaabc\right):$

The number which is repeated thrice can be chosen in 3 ways {from 1, 2, and 3}

The number $aaabc$ can be rearranged in $\frac{5!}{3!}=\frac{120}{6}=20$ ways.

There can be $3\times 20=60$ such numbers.

$Case-2\left(aabbc\right):$

The number which is not repeated can be chosen in 3 ways {from 1, 2, and 3}

The number $aabbc$ can be rearranged in $\frac{5!}{2!\times 2!}=\frac{120}{4}=30$ ways

There can be $3\times 30=90$ such numbers.

Number of 5-digit numbers that follow the given restrictions $=60+90=150$