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If ${}^n{C_x}\, = \,{}^n{C_y}\,$, then $x + y\, = \,$

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Last updated date: 13th Jun 2024
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Answer
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Hint: To calculate the value of $x + y$ we can compare this given equation with the standard equation of combination in terms of $n$ and $r$, then by comparing we will get the respective values for $x$ and $y$ and then finally by substituting those values of $x$ and $y$ in $x + y$ we will get the value of $x + y$.

Formula used: ${}^n{C_r}\, = \,{}^n{C_{n - r}}$ where $n$ is total items and $r$ is selected number of items.

Complete step-by-step answer:
According to question we are given,
${}^n{C_x}\, = \,{}^n{C_y}$ and we have to find the value of $x + y$
As we know the property of Combination which says ${}^n{C_r}\, = \,{}^n{C_{n - r}}$
So, comparing both of the equations that is,
${}^n{C_x}\, = \,{}^n{C_y}\,$ and ${}^n{C_r}\, = \,{}^n{C_{n - r}}$ we get
$r = x$ and $n - r\, = \,y$
Now, by comparing both the equations we got the value of $x$ and $y$ as $r$ and $n - r$ respectively.
So, now we can find the value of $x + y$as,
Substituting values of $x$ and $y$ in $x + y$ we get,
$x + y\, = \,r\, + \,n - r$
$ \Rightarrow \,x + y\, = \,n$

Thus, we get the value of $x + y$ as $n$

Additional Information: Combination is the selection of required number of things from the total number of things, without considering the order of the things. And when the order is important we use permutation. Both of these permutation and combination have different formulas and they are used as per the condition mentioned. The formula to calculate the combination is given as ${}^n{C_r}$ where $n$ is the total number of things and out of it $r$ things are selected.
And combination formula is given as
$C\left( {n,r} \right)\, = \,\dfrac{{n!}}{{r!\left( {n - r} \right)!}}$ (! Symbol is the factorial symbol )
For ex, $3!\, = \,3 \times 2 \times 1\, = \,6$
Also if ${}^n{C_r} = \,{}^n{C_y}$ then we can say $x = y$ or $x + y = n$

Note: Be careful while comparing the value of $x$ and $y$ with the desired property of combination, that is ${}^n{C_r} = {}^n{C_{n - r}}$. There are various other properties of combination, we have to choose them according to the given condition in question. Also after finding the values to $x$ and $y$ , don’t leave the question there, as we are asked to find their sum.