Question

Consider the following statements:
I: The number of positive integral solutions of $x_1+x_2+x_3+x_4=10$ is $286$
II: If $25!=10^n\times k,(k\in N)$ then $n=6$

• A. Only I is true
• B. Only II is true
• C. Both I and II are true
• D. Both I and II are false

Answer 1

Answer: (B)

The number of positive integral solution of
$x_1+x_2+x_3+x_4=10$ is
${}^9{C}_3=\frac{9\times 8\times 7}{1\times 2\times 3}=84$
$\therefore$ Statement $I$ is false.
The exponent of 2 in $25!$ is
$\left[\frac{25}{2}\right]+\left[\frac{25}{4}\right]+\left[\frac{25}{8}\right]+\left[\frac{25}{16}\right]$
$=12+6+3+1=22$
The exponent of 5 in (25)! is
$\left[\frac{25}{5}\right]+\left[\frac{25}{25}\right]=5+1=6$
$\therefore$ Exponent of 10 in $25!=6$
Hence, $(25)!=10^6\times k$
$\Rightarrow k\in N$
$\therefore$ Statement II is true.