The value of $r$ for which $^{15}C_{r + 3} = {}^{15}C_{2r - 6}$ is:

In how many different ways can the letters of the word '$DETAIL$' be arranged in such a way that the vowels occupy only the odd positions?

In how many different ways can the letters of the word '$LEADING$' be arranged in such a way that the vowels always come together?

From a group of $7$ men and $6$ women,five persons are to be selected to form a committee so that at least $3$ men are there on the committee. In how many ways can it be done?

From $8$ men and $6$ women,a group of seven persons is to be formed. In how many ways can the group be formed such that the group includes at least two ladies?

Given $6$ different letters of an alphabet. Words with $4$ letters are formed from these given letters. The number of words which have at least one letter repeated and no two same letters are together,is: