Question

How many integral solutions are there to a + b + c = 18 where $a\ge 1,b\ge 2,andc\ge 3$?

• A. ${}^{19}{C}_3$
• B. ${}^{17}{C}_2$
• C. ${}^{15}{C}_3$
• D. ${}^{14}{C}_2$

Answer 1

The correct option is D ${}^{14}{C}_2$

Let $u\ge 0,v\ge 0,w\ge 0$, then
$a\ge u+1,b\ge v+2,c\ge w+3$
$\therefore$ a + b + c = 18
$\Rightarrow$ u + 1 + v + 2 + w + 3 = 18
$\Rightarrow$ u + v + w = 12
$\therefore$ Total number of solutions of this equation is
${}^{12+3-1}{C}_{3-1}{=}^{14}{C}_2$