Question

If the expression is $^{n^2-n}C_2 =^{n^2-n}C_{10}$, then find the value of $n$.
A. 12
B. 4 only.
C. -3 only.
D. 4 or -3

Answer 1

Hint: To find value of $n$ , first compare the expression with the formula. Find a quadratic equation. Solve the quadratic equation for value of $n$ .

Formula used: $^{n}C_a =^{n}C_b \Rightarrow n-b=a$

Complete step by step solution: Given expression is $^{n^2-n}C_2 =^{n^2-n}C_{10}$.
Compare the expression with $^{n}C_a =^{n}C_b$
$^{n^2-n}C_2 =^{n^2-n}C_{10} \Leftrightarrow ^{n}C_a =^{n}C_b$
$\Rightarrow a=2$
$\Rightarrow b=10$

Apply the formula to find value of $n$ .
$^{n}C_a =^{n}C_b$
$n-b =a$
$^{n^2-n}C_2 =^{n^2-n}C_{10}$
$n^2-n-10=2$

Here, simplifies the equation by subtracting 2 on both sides of the equation.
_{$n^2-n-10-2=2-2$
$n^2-n-12=0$}
Find the factors of the equation.
_{$n^2-n-12=0$
$n^2-4n+3n-12=0$}
Take common factors.
_{$n^2-4n+3n-12=0$
$n(n-4) + 3(n-4) = 0$
$(n-4)(n+3)=0$}
Now equate the brackets to zero.
_{$(n-4)=0$
$(n+3)=0$}
Now solve the equation for $n$ , add 4 on both sides.
_{$n-4+4=0+4$
$n=4$}
Also isolate n from $(n+3)=0 $by subtracting 3 on both sides.
$n+3 -3 =0-3$
$n=-3$
Hence, the value of $n$ is either 4 or -3.


Thus, Option (D) is correct.

Note:The mistake happen by students is considering value of $n$ either 2 or 10 from expression $^{n^2-n}C_2 =^{n^2-n}C_{10}$. This is the wrong method.