If ${ }^{2 n} C_{r} = { }^{2 n} C_{r+2}$,find $r$.

In how many ways can a team of $11$ players be formed out of $25$ players,if $6$ out of them are always to be included and $5$ are always to be excluded?

$A$ person wants to climb a $n$-step staircase using one step or two steps. Let $C_n$ denote the number of ways of climbing the $n$-step staircase. Then $C_{18} + C_{19}$ equals

The number of arrangements of the letters of the word $PALANHAR$ in which no two vowels are together and exactly two vowels are at odd places,is

In how many different ways can the letters of the word '$BANKING$' be arranged?

In how many ways can $5$ keys be put in a ring?