Question

Number of ways of creating a $5$ digit number with the help of $1,2$ and $3$ in such a way that sum of digits is $8$

• A. $32$
• B. $28$
• C. $36$
• D. $30$

Answer 1

The correct option is D $30$

Apply multinomial theroem because maximum value of a digit is $3$.
We need to find coefficient of $x^8$ in :$(x^1+x^2+x^3)(x^1+x^2+x^3)(x^1+x^2+x^3)(x^1+x^2+x^3)(x^1+x^2+x^3)$
$=$ coefficient of $x^8$ in $(x+x^2+x^3{)}^5$
$=$ coefficient of $x^8$ in $x^5(1+x+x^2{)}^5$
$=$ coefficient of $x^3$ in $(1+x+x^2{)}^5$
$=$ coefficient of $x^3$ in ${\left(\frac{1-x^3}{1-x}\right)}^5$
$=$ coefficient of $x^3$ in $(1-x^3{)}^5(1-x{)}^{-5}$
$=$ coefficient of $x^3$ in $(1-{}^5{C}_1{x}^3+\cdots )(1-x{)}^{-5}$
$={}^{5+3-1}{C}_3-{}^5{C}_1{}^{5+0-1}{C}_0$
$={}^7{C}_3-5{}^4{C}_0$
$=35-5$
$=30$