If \({ }^n C_r\) denotes the number of combinations of ' \(n\) ' things taken ' \(r\) ' at a time, then the…

If \({ }^n C_r\) denotes the number of combinations of ' \(n\) ' things taken ' \(r\) ' at a time, then the expression \({ }^n C_{r+1}+{ }^n C_{r-1}+2^n C_r\) equals
  1. \({ }^{n+2} C_r\)
  2. \({ }^{n+2} C_{r+1}\)
  3. \({ }^{n+1} C_r\)
  4. \({ }^{n+1} C_{r+1}\)

Solution

\(\begin{aligned} { }^n C_{r+1} & +{ }^n C_{r-1}+2^n C_r \\ & =\left({ }^n C_r+{ }^n C_{r+1}\right)+\left({ }^n C_r+{ }^n C_{r-1}\right) \\ & ={ }^{n+1} C_{r+1}+{ }^{n+1} C_r={ }^{n+2} C_{r+1} \end{aligned}\) Hence, option (b) is correct.

Asked in: AP EAMCET 2020 (21 Sep Shift 1)

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