The number of permutations of n different objects, taken r at a line, when repetitions are allowed, is ______.

The number of different words that can be formed from the letters of the word INTERMEDIATE such that two vowels never come together is ______.

Three balls are drawn from a bag containing 5 red, 4 white and 3 black balls.The number of ways in which this can be done if at least 2 are red is ______

The number of six-digit numbers, all digits of which are odd is ______.

In a football championship, 153 matches were played . Every two teams played one match with each other. The number of teams, participating in the championship
is ______.

The total number of ways in which six ‘+’ and four ‘–’ signs can be arranged in a line such that no two signs ‘–’ occur together is ______.

A committee of 6 is to be chosen from 10 men and 7 women so as to contain atleast 3 men and 2 women. In how many different ways can this be done if two particular women refuse to serve on the same committee.

A box contains 2 white balls, 3 black balls and 4 red balls. The number of ways  threeballs be drawn from the box if at least one black ball is to be included in the draw is ______.

Three letters can be posted in five letterboxes in 3^5 ways.

In a steamer there are stalls for 12 animals, and there are horses, cows and calves (not less than 12 each) ready to be shipped. They can be loaded in 3^12  ways.

If some or all of n objects are taken at a time, the number of combinations
is 2^n–1.