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.