Permutations and Combinations

The value of the expression \(\,{}^{47}{C_4} + \sum\limits_{j = 1}^5 {^{52 - j}\,{C_3}}\) is equal to

In a certain test, \({a_i}\) students gave wrong answers to atleast i questions, where i = 1, 2,..., k. No student gave more than k wrong answers. The total number of wrong answers given is.....................................

Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the numbers of words which have at least one letter repeated are

Five balls of different colours are to be placed in there boxes of different size. Each box can hold all five. In how many different ways can be place the balls so that no box remains emply?

\({}^n{C_{r - 1}} = 36,{}^n{C_r} = 84\,\,and\,\,{}^n{C_{r + 1}} = 126\), then r is :

Six X' s have to be placed in the squares of figure below in such a way that each row contains at least one X. In how many different ways can this be done.