If \(\rm \begin{pmatrix} 15\\ 8\end{pmatrix}+\begin{pmatrix} 15\\ 7\end{pmatrix}=\begin{pmatrix} \rm n\\ \rm r\end{pmatrix}\)

then the values of n and r are:

Where, \(\begin{pmatrix} \rm n\\ \rm r\end{pmatrix} = {}^nC_r\)

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  1. 16 and 7.
  2. 16 and 8.
  3. 16 and 9.
  4. 30 and 15.

Answer (Detailed Solution Below)

Option 2 : 16 and 8.
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Detailed Solution

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Concept:

  • nCr = nCn-r.
  • If nCx = nCy, then x = y or x + y = n.
  • nCr + nCr-1 = n+1Cr.

 

Calculations:

\(\rm \begin{pmatrix} 15\\ 8\end{pmatrix}+\begin{pmatrix} 15\\ 7\end{pmatrix}=\begin{pmatrix} \rm n\\ \rm r\end{pmatrix}\)

⇒ 15C8 + 15C7 = nCr

Using nCr + nCr-1 = n+1Cr:

⇒ ​15C8 + 15C8-1 = 15+1C8nCr

⇒ ​16C8 = nCr

n = 16 and r = 8.

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