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Chapter 5: Problem 417

Ten different letters of an alphabet are given Words with 5 letters are formed from the given letters. The no, of words which have at least one letter repeated is (a) 69760 (b) 30240 (c) 9948 (d) 10680

Short Answer

Expert verified
The number of 5-letter words with at least one letter repeated is 69760 (Option a).

Step by step solution

01

Calculate total number of 5-letter words without any restrictions

To form a 5-letter word, each position can be filled by any of the 10 available letters. Therefore, there are 10 choices for the first letter, 10 choices for the second letter, and so on. Using the multiplication principle, the total number of 5-letter words without any restrictions is \(10 \times 10 \times 10 \times 10 \times 10\), which is equal to \(10^5\).
02

Calculate the number of 5-letter words with no repeated letters

To form a 5-letter word with no repeated letters, each position must be filled by a different letter. There are 10 choices for the first letter. Since we cannot repeat it for the second letter, there are 9 choices left for the second letter, 8 choices left for the third letter, 7 choices left for the fourth letter, and 6 choices left for the fifth letter. Using the multiplication principle, the total number of 5-letter words with no repeated letters is \(10 \times 9 \times 8 \times 7 \times 6\).
03

Apply the Inclusion-Exclusion principle

According to the Inclusion-Exclusion principle, the number of 5-letter words with at least one letter repeated is the difference between the total number of 5-letter words and the number of 5-letter words with no repeated letters. Thus, the number of words with at least one letter repeated is \(10^5 - (10 \times 9 \times 8 \times 7 \times 6)\).
04

Calculate the final result

Now we can calculate the difference to find the number of 5-letter words with at least one letter repeated: \[10^5 - (10 \times 9 \times 8 \times 7 \times 6) = 100000 - 30240 = 69760\] Therefore, the correct answer is (a) 69760.

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Most popular questions from this chapter

In a certain test there are n questions. In this test \(2^{\mathrm{k}}\) students gave wrong answers to at least \((\mathrm{n}-\mathrm{k})\) question. \(\mathrm{k}=0,1,2 \ldots \mathrm{n} .\) If the no. of wrong answers is 4095 then value of \(\mathrm{n}\) is (a) 11 (b) 12 (c) 13 (d) 15

The reminder when number \(1 !+2 !+3 !+4 !+\ldots \ldots+100 !\) is divided by 240 is (a) 153 (b) 33 (c) 73 (d) 187

A car will hold 2 in the front seat and 1 in the rear seat. If among 6 persons 2 can drive, than number of ways in which the car can be filled is (a) 10 (b) 20 (c) 30 (d) 40

The product of n natural number \(\mathrm{n} \geq 2\) is (a) not divisible by n (b) divisible by \(\mathrm{n}\), but not by \(2 \mathrm{n}\) (c) divisible by \(2 \mathrm{n}\), but not by n! (d) divisible by n!

Nine hundred distinct n digit numbers are to be formed using only the 3 digits \(2,5,7 .\) The smallest value of \(\mathrm{n}\) for which this is possible is (a) 6 (b) 7 (c) 8 (d) 9
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