Question

If \[{}^8{C_r} = {}^8{C_{r + 2}}\], then find the value of \[{}^r{C_2}\].
A. \[8\]
B. \[3\]
C. \[5\]
D. \[2\]

Answer 1

Hint: In the given question, we need to find the value of \[{}^r{C_2}\]. For this, we will simplify the expression \[{}^8{C_r} = {}^8{C_{r + 2}}\] using combination formula to get the desired result.



Formula Used: The following formula used for solving the given question.
The formula of combination is given by, \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\]
Here, \[n\] is total number of things and \[r\] is number of things need to be selected from total things.



Complete step by step solution:We know that \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\]
Here, \[n\] is total number of things and \[r\] is number of things need to be selected from total things.
Now, we will simplify the expression \[{}^8{C_r} = {}^8{C_{r + 2}}\].
 \[\dfrac{{8!}}{{r!\left( {8 - r} \right)!}} = \dfrac{{8!}}{{(r + 2)!\left( {8 - (r + 2)} \right)!}}\]
By simplifying, we get
\[\dfrac{1}{{r!\left( {8 - r} \right)!}} = \dfrac{1}{{(r + 2)!\left( {8 - (r + 2)} \right)!}}\]
This gives \[(r + 2)!\left( {6 - r} \right)! = r!\left( {8 - r} \right)!\]
By simplifying, we get
 \[\left[ {(r + 2)(r + 1)(r)!} \right]\left( {6 - r} \right)! = r!\left[ {\left( {8 - r} \right)(7 - r)(6 - r)!} \right]\]
By simplifying further, we get
\[(r + 2)(r + 1) = \left( {8 - r} \right)(7 - r)\]
\[{r^2} + r + 2r + 2 = 56 - 8r - 7r + {r^2}\]
So, we get \[3r + 15r = 54\]
\[18r = 54\]
By simplifying, we get
\[r = 3\]
Now, the value of \[{}^r{C_2}\]is given by
\[{}^r{C_2} = {}^3{C_2} = \dfrac{{3!}}{{2!\left( {3 - 2} \right)!}}\]
By simplifying, we get
\[{}^r{C_2} = {}^3{C_2} = 3\]
Hence, if \[{}^8{C_r} = {}^8{C_{r + 2}}\], then the value of \[{}^r{C_2}\] is \[3\].



Option ‘B’ is correct

Additional Information:Combinations are ways to choose elements from a group in mathematics in which the order of the selection is irrelevant. Suppose we have a three numbers. So, combination determines how many ways through which we can choose two numbers from each group.



Note: Many students make mistake in calculation part as well as writing the combination rule. This is the only way, through which we can solve the example in simplest way. Also, it is essential to find the appropriate value of \[r\]as the final result is totally depend on the value of \[r\].