4 Answers
49\frac{n!}{r!(n-r)!}+\frac{n!}{(r-1)!(n-r+1)!}\\ =\frac{n!}{(r-1)!(n-r)!}\left[ \frac{1}{r}+\frac{1}{n-r+1}\right]\\ =\frac{n!}{(r-1)!(n-r)!}\left[\frac{n+1}{(r(n-r+1)} \right]\\ =\frac{(n+1)!}{r!(n-r+1)!}
30Convert the statement to:
\binom{n}{r} + \binom{n}{r-1}=\binom{n+1}{r}
Now, think of the this situation:
You have n+1 players out of which one player has a slight niggle. You have to form a team of r players. LHS represents the no. of ways of forming the team, where you take the r players from the n fit players or take the slightly injured player and choose the remaining r-1 players from the fit n players.
Now, RHS also says the same thing.. We want to choose r players from a set of n+1 players. Hence, proved :)