Question

If \[{}^{15}{C_{3r}} = {}^{15}{C_{r + 3}}\], then the value of r is:
(A) 3
(B) 4
(C) 5
(D) 6

Answer 1

Hint: If \[{}^{15}{C_{3r}} = {}^{15}{C_{r + 3}}\] the value of r+3 and 3r is equal so by equalizing them get the value of r but from this we get the value of r as a rational number so use the formula \[{}^n{C_r} = {}^n{C_{n - r}}\] and convert \[{}^{15}{C_{3r}}\]to \[{}^{15}{C_{15 - 3r}}\] and then compare the value of 15-3r and r+3 to get the value of r.

Complete step-by-step answer:
\[{}^{15}{C_{3r}} = {}^{15}{C_{r + 3}}\]
From above equation the value of the value of 3r and r+3
\[3r = r + 3\]
\[2r = 3\]
\[r = \dfrac{3}{2}\]
r is integer number so convert \[{}^{15}{C_{3r}}\] to \[{}^{15}{C_{15 - 3r}}\] by using formula \[{}^n{C_r} = {}^n{C_{n - r}}\] and compare \[15 - 3r\] and \[r + 3\].
\[15 - 3r = r + 3\]
\[15 - 3 = r + 3r\]
\[4r = 12\]
\[r = 3\]
So,option A is the required answer.

Note: Another approach to solve this question is by using this formula:
\[{}^n{C_r} = {}^n{C_s}\]
Where \[r = s\] or \[r + s = n\]
So, for this question
\[{}^{15}{C_{3r}} = {}^{15}{C_{r + 3}}\]
\[3r = r + 3\]
\[2r = 3\]
\[r = \dfrac{3}{2}\]
\[\begin{array}{l}
r + s = n\\
3r + r + 3 = 15\\
4r = 12\\
r = 3
\end{array}\]
So, we get the two values of r are \[3\] and \[\dfrac{3}{2}\]. Since r has to be integer so the value of r is 3.