# 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

Answer

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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.

Last updated date: 27th Sep 2023

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