# Ten different letters of the alphabet are given. Words with five letters are formed from these given letters. Then, the number of words which have at least one letter repeated is $69760$ .

A.True

B.False

Last updated date: 26th Mar 2023

•

Total views: 306.6k

•

Views today: 2.83k

Answer

Verified

306.6k+ views

Hint: The given problem is related to permutation and combinations. Try to recall the formulae related to arrangement of letters with and without repetition.

Complete step-by-step answer:

It is given that five letter words are formed from ten different letters of the alphabet. We need to find the number of words that have at least one letter repeated. Instead of directly finding the number of words that have at least one letter repeated, we will find the total number of words that can be formed and the number of words that can be formed with no letter repeated. The difference of both will be the number of words that have at least one letter repeated.

Now, we will consider the word as five blank spaces. To find the total number of words that can be formed, we have to consider the fact that each blank can be filled in $10$ ways. So, total number of words that can be formed is equal to $10\times 10\times 10\times 10\times 10=100000$ . Hence, a total of $100000$ five letter words can be formed from ten different letters of the alphabet.

Now, in case of no repetition, after filling the first blank, there will be $9$ possible ways of filling the second blank. Then, after filling the second blank, there will be $8$ possible ways of filling the third blank. Then, after filling the third blank, there will be $7$ possible ways of filling the fourth blank. Then, after filling the fourth blank, there will be $6$ possible ways of filling the fifth blank. So, the number of ways in which a five-letter word can be formed with no letter repeated is given as $10\times 9\times 8\times 7\times 6=30240$ .

Now, the number of ways in which a five-letter word can be formed with at least one letter repeated is given as $100000-30240=69760$ .

Hence, option A. “True” is the correct answer.

Note: Calculation mistakes are very common in questions related to permutation and combination. Be careful with the calculations where multiplication, addition or subtraction of large numbers is to be done.

Complete step-by-step answer:

It is given that five letter words are formed from ten different letters of the alphabet. We need to find the number of words that have at least one letter repeated. Instead of directly finding the number of words that have at least one letter repeated, we will find the total number of words that can be formed and the number of words that can be formed with no letter repeated. The difference of both will be the number of words that have at least one letter repeated.

Now, we will consider the word as five blank spaces. To find the total number of words that can be formed, we have to consider the fact that each blank can be filled in $10$ ways. So, total number of words that can be formed is equal to $10\times 10\times 10\times 10\times 10=100000$ . Hence, a total of $100000$ five letter words can be formed from ten different letters of the alphabet.

Now, in case of no repetition, after filling the first blank, there will be $9$ possible ways of filling the second blank. Then, after filling the second blank, there will be $8$ possible ways of filling the third blank. Then, after filling the third blank, there will be $7$ possible ways of filling the fourth blank. Then, after filling the fourth blank, there will be $6$ possible ways of filling the fifth blank. So, the number of ways in which a five-letter word can be formed with no letter repeated is given as $10\times 9\times 8\times 7\times 6=30240$ .

Now, the number of ways in which a five-letter word can be formed with at least one letter repeated is given as $100000-30240=69760$ .

Hence, option A. “True” is the correct answer.

Note: Calculation mistakes are very common in questions related to permutation and combination. Be careful with the calculations where multiplication, addition or subtraction of large numbers is to be done.

Recently Updated Pages

Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE