If $^{10}C_r = ^{10}C_{r+2}$,then $^5C_r$ equals:

The number of ways in which the letters of the word $ARRANGE$ can be arranged such that both $R$ do not come together is

The sum $\sum_{r=1}^{10} (r^2 + 1) \times (r!)$ is equal to

In how many ways can the letters of the word '$NUTAN$' be arranged when consonants occupy odd places and vowels come on even places?

$A$ committee of $5$ members is to be formed out of $3$ trainees,$4$ professors,and $6$ research associates. In how many different ways can this be done if the committee should have all $4$ professors and $1$ research associate $OR$ all $3$ trainees and $2$ professors?

How many numbers lying between $99$ and $1000$ can be formed using the digits $2, 3, 7, 0, 8, 6$,if each digit can be used only once in each number?