Question:
Write the expression nCr +1 + nCr − 1 + 2 × nCr in the simplest form.
Solution:
${ }^{n} C_{r+1}+{ }^{n} C_{r-1}+2 \cdot{ }^{n} C_{r}$
$=\left({ }^{n} C_{r+1}+{ }^{n} C_{r}\right)+\left({ }^{n} C_{r}+{ }^{n} C_{r-1}\right) \quad\left[{ }^{n} C_{r}+{ }^{n} C_{r-1}={ }^{n+1} C_{r}\right]$
$={ }^{n+1} C_{r+1}+{ }^{n+1} C_{r} \quad\left[{ }^{n} C_{r}+{ }^{n} C_{r-1}={ }^{n+1} C_{r}\right]$
$={ }^{n+2} C_{r+1}$
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