Question

Show that n+1C = nCr-1 + nr.

Answer 1

Proof:

We have to show that

`\binom{n}{r-1}+\binom{n}{r}=\binom{n+1}{r}`Solving the LHS of the above relation we have
`\\\\(n!)/((n-(r-1))!\cdot (r-1)!)+(n!)/((n-r)!\cdot r!)\\\\(n!)/((n-r+1)!\cdot (r-1)!)+(n!)/((n-r)!\cdot r!)\\\\(n!)/((n-r+1)!\cdot (r-1)!)* (r)/(r)+(n!)/((n-r)!\cdot r!)* (n-r+1)/(n-r+1)\\\\(rn!)/((n-r+1)!\cdot r!)+((n-r+1)\cdot n!)/((n-r+1)!\cdot r!)\\\\(n!)/((n-r+1)!\cdot r!)\cdot (r+n-r+1)\\\\((n+1)!)/((n-r+1)!* r!)=\binom{n+1}{r}`

Hence proved