If $^{43}C_{r-6} = ^{43}C_{3r+1}$,then the value of $r$ is

$A$ library has $a$ copies of one book,$b$ copies of each of two books,$c$ copies of each of three books,and single copies of $d$ books. The total number of ways in which these books can be arranged is

The sum of all $4$ digit numbers that can be formed by using the digits $2, 4, 6, 8$ (repetition of digits not allowed) is

The number of positive integral solutions of the equation $xyz = 90$ is equal to:

The number of arrangements of the letters of the word $BANANA$ in which two $N$s do not appear adjacently is

In how many ways can five examination papers be arranged so that physics and chemistry papers never come together?