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

The Correct Option is C

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 5C(x,3)=5C(3,3), which leads to x=5−3=2.

Hence, the value of x is 2.