Question

The number of integral solutions for the equation a + b + c + d = 12, where (a, b, c, d) $\ge$ - 1 is:

• A. ${}^{19}{C}_3$
• B. ${}^{18}{C}_4$
• C. ${}^{20}{C}_4$
• D. None of these

Answer 1

The correct option is A ${}^{19}{C}_3$

$\because a\ge -1,b\ge -1,c\ge -1,d\ge -1$
Let $u\ge 0,v\ge 0,w\ge 0,x\ge 0$
$\therefore$ a + b + c + d = 12
$\Rightarrow$ (u - 1) + (v - 1) + (w - 1) + (x - 1) = 12
$\Rightarrow$ u + v + w + x = 16
$\therefore$ Required number of solutions ${=}^{16+4-1}{C}_{4-1}{=}^{19}{C}_3$