Question

If the number of integral solutions of $x+y+z+w<25$ are ${}^{23}{C}_{\lambda }^{}$ , such that $x>-2,y>1,z\ge 0,w>3$ , 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 $a>0$
$\Rightarrow \left(x+1\right)+\left(y-2\right)+z+\left(w-4\right)+\left(a-1\right)$
$=t_1+t_2+t_3+t_4+t_5=25+1-2-4-1=19$
$x>-2\Rightarrow x\ge -1\Rightarrow x+1\ge 0\Rightarrow t_1\ge 0,y>1\Rightarrow y\ge 2\Rightarrow y-2\ge 0\Rightarrow t_2\ge 0$
$z\ge 0\Rightarrow t_3\ge 0$
$w>3\Rightarrow w\ge 4\Rightarrow w-4\ge 0\Rightarrow t_4\ge 0$
$a>0\Rightarrow a\ge 1\Rightarrow a-1\ge 0\Rightarrow t_5\ge 0$
$t_1+t_2+t_3+t_4+t_5=19$ where $t_i\ge 0$
Number of integral solutions $={}^{19+5-1}{C}_{5-1}={}^{23}{C}_4={}^{23}{C}_{19}$