ncert books for class maths cbsc · Part

A Note If the repetition of the letters was allowed, how many words can be formed? One can easily understand that each of the vacant places can be filled in succession in different ways. Hence, the required number of words = × × × = . Example Given flags of different colours, how many different signals can be generated, if a signal requires the use of flags one below the other?

Solution There will be as many signals as there are ways of filling in vacant places in succession by the flags of different colours. The upper vacant place can be filled in different ways by anyone of the flags; following which, the lower vacant place can be filled in different ways by anyone of the remaining different flags. Hence, by the multiplication principle, the required number of signals = × = . Example How many digit even numbers can be formed from the digits , , , , if the digits can be repeated?

Solution There will be as many ways as there are ways of filling vacant places in succession by the five given digits. Here, in this case, we start filling in unit’s place, because the options for this place are and only and this can be done in ways; following which the ten’s place can be filled by any of the digits in different ways as the digits can be repeated. Therefore, by the multiplication principle, the required number of two digits even numbers is × , i.e., . Example Find the number of different signals that can be generated by arranging at least flags in order (one below the other) on a vertical staff, if five different flags are available.

Solution A signal can consist of either flags, flags, flags or flags. Now, let us count the possible number of signals consisting of flags, flags, flags and flags separately and then add the respective numbers. There will be as many flag signals as there are ways of filling in vacant places in succession by the flags available. By Multiplication rule, the number of ways is × = .

Similarly, there will be as many flag signals as there are ways of filling in vacant places in succession by the flags. MATHEMATICS The number of ways is × × = . Continuing the same way, we find that The number of flag signals = × × × = and the number of flag signals = × × × × = Therefore, the required no of signals = + + + = . EXERCISE .

. How many -digit numbers can be formed from the digits , , , and assuming that repetition of the digits is allowed? repetition of the digits is not allowed? .

How many -digit even numbers can be formed from the digits , , , , , if the digits can be repeated? . How many -letter code can be formed using the first letters of the English alphabet, if no letter can be repeated? .

How many -digit telephone numbers can be constructed using the digits to if each number starts with and no digit appears more than once? . A coin is tossed times and the outcomes are recorded. How many possible outcomes are there?

. Given flags of different colours, how many different signals can be generated if each signal requires the use of flags, one below the other? . Permutations In Example of the previous Section, we are actually counting the different possible arrangements of the letters such as ROSE, REOS, ..., etc.

Here, in this list, each arrangement is different from other. In other words, the order of writing the letters is important. Each arrangement is called a permutation of 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 r is cumbersome and we need a notation which will help to reduce the size of this expression.

The symbol n ! (read as factorial n or n factorial ) comes to our rescue. In the following text we will learn what actually n ! means.

. . Factorial notation The notation n ! represents the product of first n natural numbers, i.e., the product × × × .

. . × ( n – ) × n is denoted as n !. We read this symbol as ‘ n factorial’.

and so on. We define ! = We can write ! = × !

= n ( n – ) ! = n ( n – ) ( n – ) ! [provided ( n ≥ )] = n ( n – ) ( n – ) ( n – ) ! [provided ( n ≥ )] and so on.

MATHEMATICS Example Evaluate (i) ! (ii) ! (iii) ! – !

= – = . Example Compute (i) ! !