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Chapter 5: Problem 368

Ten different letters of english alphabet are given. Out of these letters, words of 5 letters are formed. How many words are formed when at least one letter is repeated? (a) 69760 (b) 98748 (c) 96747 (d) 97147

Short Answer

Expert verified
The number of 5-letter words with at least one letter repeated is 69,760.

Step by step solution

01

Calculate total number of 5-letter words without any restrictions

First, let's calculate the number of ways we can form 5-letter words without any restrictions. In such a case, we have 10 choices for the first letter, 10 choices for the second letter, 10 choices for the third letter, 10 choices for the fourth letter, and 10 choices for the fifth letter. Using the counting principle, the total number of 5-letter words without any restrictions is: \(10 \times 10 \times 10 \times 10 \times 10 = 10^5 = 100000\)
02

Calculate the total number of 5-letter words without any repetitions

In this case, we have 10 choices for the first letter, 9 choices for the second letter, 8 choices for the third letter, 7 choices for the fourth letter, and 6 choices for the fifth letter. Using the counting principle, the total number of 5-letter words without any repetitions is: \(10 \times 9 \times 8 \times 7 \times 6 = 30240\)
03

Calculate the number of words with at least one letter repeated

Now, we know the total number of words with no restrictions and the total number of words without any repetitions. To find the number of words with at least one letter repeated, we can subtract the number of words without any repetitions from the total number of words without any restrictions: Number of 5-letter words with at least one letter repeated = Total number of 5-letter words without any restrictions - Total number of 5-letter words without any repetitions \(= 100000 - 30240 = 69760\)
04

Identify the correct choice

The number of 5-letter words with at least one letter repeated is 69760. So the correct choice is: (a) 69760

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Most popular questions from this chapter

Three boys and three girls are to be seated around a round table in a circle. Among them the boy \(\mathrm{X}\) does not want any girl neighbour and the girl \(\mathrm{Y}\) does not want any boy neighbour then the no. of arrangement is (a) 2 (b) 4 (c) 23 (d) 33

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See all solutions

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