There are $15$ buses running between Delhi and Mumbai. In how many ways can a man go to Mumbai and return by a different bus?

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

$A$ lady gives a dinner party for six guests. The number of ways in which they may be selected from among ten friends,if two of the friends will not attend the party together is

The total number of seven-digit numbers the sum of whose digits is even is

The number of arrangements that can be formed from the letters $a, b, c, d, e, f$ taken $3$ at a time without repetition and each arrangement containing at least one vowel,is

The number of $3$-digit odd numbers that can be formed by using the digits $1, 2, 3, 4, 5, 6$ when the repetition is allowed,is