Question

If $4$ dice are rolled once, the number of ways of getting the sum as $10$ is

• A. $64$
• B. $80$
• C. $84$
• D. $126$

Answer 1

The correct option is B $80$

Here, $x_1+x_2+x_3+x_4=10$
$6\ge x_1,x_2,x_3,x_4\ge 1$
Since the minimum value of every dice will be $1$
$\therefore X_1+1+X_2+1+X_3+1+X_4+1=10\Rightarrow X_1+X_2+X_3+X_4=6$
Where $5\ge X_1,X_2,X_3,X_4\ge 0$
Total ways $={}^{6+4-1}{C}_{4-1}={}^9{C}_3=84$
Now subtracting those cases when any $X_i=6$
There will be $4$ cases
$\therefore$ Required number of ways
$=84-4=80$