Question

The number of solution of $x_1+x_2+x_3=51$ $(x_1,x_2,x_3)$ being odd natural numbers) is:

• A. $300$
• B. $325$
• C. $330$
• D. $350$

Answer 1

Answer: (B)

$325$

Let $x_1=2m_1+1$ $m_1\ge 0$

$x_2=2m_2+1$ $m_2\ge 0$

$x_3=2m_3+1$ $m_3\ge 0$

$\therefore x_1+x_2+x_3=2(m_1+m_2+m_3)+3$

$\therefore 2(m_1+m_2+m_3)=48$

$\therefore m_1+m_2+m_3=24$

$\therefore$ No. of integer solutions = $n+r-1C_{r-1}$

$=24+3-1C_{3-1}$

$=26C_2=\frac{26!}{24!2!}$

$=\frac{25\times 26}{2}$

$=325$