Question

If $^n$C$_2$ + $^n$C$_3$ = $^6$C$_3$ and $^n$C$_x$ = $^n$C$_3$, x ? 3, then the value of x is equal to

• A. 5
• B. 4
• C. 2
• D. 6

Answer 1

Answer: (C)

2

Answer 2

The correct option is (C): 2

Given, ${ }^{n} C_{2}+{ }^{n} C_{3}={ }^{6} C_{3}$
$\Rightarrow n=5 \left[\because{ }^{n} C_{r}+{ }^{n} C_{r-1}={ }^{n+1} C_{r}\right]$
and ${ }^{n} C_{x}={ }^{n} C_{3} \Rightarrow{ }^{5} C_{x}={ }^{5} C_{3}$
$\Rightarrow x =5-3=2$
$\left[\because\right.$ If ${ }^{n} C_{r 1}={ }^{n} C_{12} \Rightarrow$ either $r_{1}=r_{2}$ or $\left.n=r_{1}+r_{2}\right]$

The provided equations are:

1. C(n ,2)+C(n ,3)=C(n ,3)
2. C(n ,x)=C(n ,3)

Using the property of binomial coefficients, where C(n ,r)+C(n ,r −1)=C(n +1,r), we can deduce the value of n : From equation (1), we have C(n ,2)+C(n ,1)=C(n +1,2), which simplifies to n =5 since C(n ,2)+C(n ,1)=C(n +1,2).

Therefore, n =5.

Next, applying equation (2): C(n ,x)=C(n ,3), we find that C(5,x)=C(5,3).

Solving for x , we get 5 C(x ,3)=5 C(3,3), which leads to x =5−3=2.

Hence, the value of x is 2.