different letters taken all at a time . Now, if we have to determine the number of -letter words, with or without meaning, which can be formed out of the letters of the word NUMBER, where the repetition of the letters is not allowed, we need to count the arrangements NUM, NMU, MUN, NUB, ..., etc. Here, we are counting the permutations of different letters taken at a time. The required number of words = × × = (by using multiplication principle).
If the repetition of the letters was allowed, the required number of words would be × × = . PERMUTATIONS AND COMBINATIONS Definition A permutation is an arrangement in a definite order of a number of objects taken some or all at a time. In the following sub-section, we shall obtain the formula needed to answer these questions immediately. .
. Permutations when all the objects are distinct Theorem The number of permutations of n different objects taken r at a time, where < r ≤ n and the objects do not repeat is n ( n – ) ( n – ). . .( n – r + ), which is denoted by n P r .
Proof There will be as many permutations as there are ways of filling in r vacant places . . . by ← r vacant places → the n objects.
The first place can be filled in n ways; following which, the second place can be filled in ( n – ) ways, following which the third place can be filled in ( n – ) ways,..., the r th place can be filled in ( n – ( r – )) ways. Therefore, the number of ways of filling in r vacant places in succession is n ( n – ) ( n – ) . . .
( n – ( r – )) or n ( n – ) ( n – ) ... ( n – r + ) This expression for n P