Question

How many four letter words are possible using the first 5 letters of the alphabet if letters can be repeated?

Answer 1

To solve this question, we need to use the multiplication rule of counting. We can imagine four blank spaces, since we have to form four letter words. Since the letters can be repeated, the number of ways of filling each of the four spaces will be equal to five. Finally, using the multiplication rule we can determine the total number of ways of filling all the four blank spaces and hence the number of four letter words possible.

Complete step-by-step solution:
We know that the first five letters of the alphabet are a, b, c, d, and e. Therefore, the set of the letters of the alphabet from which we can form the four letter words is \left\\{ a,b,c,d,e \right\\}.
Now, since we have to form four letter words, we can imagine four empty spaces which can be filled using the letters from the set \left\\{ a,b,c,d,e \right\\}. For filling the first empty space, we have a total of five letters, which implies that the total number of filling the first blank space is equal to five. Then, for filling the second space, we again have a total of five words, since it is mentioned in the above question that the letters can be repeated. Therefore, the number of ways for filling the second space is also equal to five. Similarly, the number of ways of filling each of the third and the fourth blank space will be equal to five.
Now, using the multiplication rule of counting we can say that the total number of ways of filling all the four blank spaces will be equal to $5\times 5\times 5\times 5$, which is equal to ${{5}^{4}}$ ways.
Hence, the required number of letter words which are possible using the first five letters of the alphabet, when the letters can be repeated is equal to ${{5}^{4}}$.

Note: Do not use the additional rule of counting for solving this question. We must understand the basic difference between the application of the addition and the multiplication rule. The addition rule is used when we have a choice, while the multiplication rule is used in the opposite case. In this question, we didn’t have any choice since we had to fill all the four places of the four letter words.