Question

If the number of integral solutions of $x+y+z+w < 25$ are ${}^{23}C_{\lambda }^{}$ , such that $x>-2,y>1,z\geq 0,w>3$ , then the value of $\lambda $ is

• A. $3$
• B. $5$
• C. $17$
• D. $19$

Answer 1

Answer: (D)

$x+y+z+w < 25$
Let, $x+y+z+w+a=25$ , such that $a>0$
$\Rightarrow \left(x + 1\right)+\left(y - 2\right)+z+\left(w - 4\right)+\left(a - 1\right)$
$=t_{1}+t_{2}+t_{3}+t_{4}+t_{5}=25+1-2-4-1=19$
$x>-2\Rightarrow x\geq -1\Rightarrow x+1\geq 0\Rightarrow t_{1}\geq 0,y>1\Rightarrow y\geq 2\Rightarrow y-2\geq 0\Rightarrow t_{2}\geq 0$
$z\geq 0\Rightarrow t_{3}\geq 0$
$w>3\Rightarrow w\geq 4\Rightarrow w-4\geq 0\Rightarrow t_{4}\geq 0$
$a>0\Rightarrow a\geq 1\Rightarrow a-1\geq 0\Rightarrow t_{5}\geq 0$
$t_{1}+t_{2}+t_{3}+t_{4}+t_{5}=19$ where $t_{i}\geq 0$
Number of integral solutions $={}^{19 + 5 - 1}C_{5 - 1}={}^{23}C_{4} = {}^{23}C_{19}$