Question

If value of ${}^n{C_x} = {}^n{C_y}$, then find the value of $x + y =$

Answer 1

Here we need to find the sum of two variables. We will first solve the given equation which includes the expression of the combination. We will use the property of combination in the given equation and then on solving the equation, we get the sum of the two variables from there.

Complete step-by-step answer:
The given equation is
${}^n{C_x} = {}^n{C_y}$ ……………… $\left( 1 \right)$
We know from the properties of combination that:-
$\Rightarrow {}^n{C_r} = {}^n{C_{n - r}}$ ………… $\left( 2 \right)$
Now, we will compare the right hand side of equation $\left( 1 \right)$ with the right hand side of equation $\left( 2 \right)$ and the left hand side of equation 1 with the left hand side of equation $\left( 2 \right)$.
$x = r$ ……….. $\left( 3 \right)$
$y = n - r$ ……… $\left( 4 \right)$
Now, we will substitute the value of $r$ obtained in the equation $\left( 4 \right)$. Therefore, we get
$y = n - x$
Now, we will add the term $x$ on both sides of the equation. So,
$\Rightarrow y + x = n - x + x$
On further simplifying the terms, we get
$\Rightarrow y + x = n$
Rewriting the equation, we get
$\Rightarrow x + y = n$
Hence, the value of $x + y$ is equal to $n$.

Note: Here we have used the property of combination to get the value of the sum of two variables. To solve this problem, we need to know the basic formulas and also the different properties of the permutation and the combination. We also need to know the basic difference between the permutation and combination to avoid any mistakes. Permutation is used when we have to find the possible arrangement of elements but the combination is used when we need to find the number of ways to select a number from the collection.