Log out Go to App
Go to App

Learning Materials

Features

Discover

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The no, of ways in which ten candidates \(\mathrm{A}_{1}, \mathrm{~A}_{2} \ldots \mathrm{A}_{10}\) can be ranked, if \(\mathrm{A}_{1}\) is always above \(\mathrm{A}_{2}\) is (a) \(2 \cdot 8 !\) (b) \(9 !\) (c) \(10 !\) (d) \(5 \cdot 9 !\)

There are 10 points in a plane out of these 6 are collinear. If \(\mathrm{N}\) is the number of triangles formed by joining these points then: (a) \(\mathrm{N} \leq 100\) (b) \(100<\mathrm{N}<140\) (c) \(140<\mathrm{N}<190\) (d) \(\mathrm{N}>190\)

The number of ways in which a committee of 3 women and 4 men be chosen from 8 women and 7 men is formed if \(\mathrm{mr}\). \(\mathrm{A}\) refuses to serve on the committee if \(\mathrm{mr}\). \(\mathrm{B}\) is a member of the committee is (a) 420 (b) 840 (c) 1540 (d) none of these

If a denotes the number of permutation of \(\mathrm{x}+2\) things taken all at a time, \(b\) the number of permutation of \(x\) things taken 11 at a time and \(c\) the number of permutation of \(x-11\) things taken all at a time such that \(\mathrm{a}=182 \mathrm{bc}\) then the value of \(\mathrm{x}\) is (a) 15 (b) 12 (c) 10 (d) 18

The number of 9 digit numbers formed using the digit 223355888 such that odd digits occupy even places is (a) 16 (b) 36 (c) 60 (d) 80
See all solutions

Recommended explanations on English Textbooks

Textual Analysis

Read Explanation

Syntax

Read Explanation

Discourse

Read Explanation

English Language Study

Read Explanation

Language Acquisition

Read Explanation

Rhetorical Analysis Essay

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free