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