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Ten different letters of an alphabet are given, words with 4 letters a

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Bunuel
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Ten different letters of an alphabet are given, words with 4 letters a [#permalink] New post   12 Mar 2021, 03:15
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73% (02:14) correct 27% (02:10) wrong based on 90 sessions
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Ten different letters of an alphabet are given, words with 4 letters are formed from these given letter. Then what is the number of words which have at least one letter repeated ?

(A) 10000
(B) 9900
(C) 9000
(D) 5040
(E) 4960
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E
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Re: Ten different letters of an alphabet are given, words with 4 letters a [#permalink] New post   18 Mar 2021, 06:35
If we pick 4 letters from 10 different letters with repetition, the number of words will be 10*10*10*10 = 10,000

If we pick 4 letters from 10 different letters without repetition, the number of words will be 10*9*8*7 = 5,040

If we exclude words without repetition from words with repetition, we will get the number of words that have at least one word repeated.

10,000 - 5,040 = 4960. (OPTION: E IMO)
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Re: Ten different letters of an alphabet are given, words with 4 letters a [#permalink] New post   29 Mar 2021, 11:10
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Bunuel wrote:
Ten different letters of an alphabet are given, words with 4 letters are formed from these given letter. Then what is the number of words which have at least one letter repeated ?

(A) 10000
(B) 9900
(C) 9000
(D) 5040
(E) 4960

Solution:

The total number of 4-letter words that can be formed is 10 x 10 x 10 x 10 = 10,000. The total number of 4-letter words without any repeating letters that can be formed is 10 x 9 x 8 x 7 = 5,040. Therefore, the total number of 4-letter words (with at least one repeating letter) that can be formed is 10,000 - 5,040 = 4,960.

Answer: E
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Re: Ten different letters of an alphabet are given, words with 4 letters a [#permalink] New post   30 Mar 2021, 07:12
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Given
    • Ten different letters of an alphabet are given.
    • Out of the given letters, words with 4 letters are to be formed.

To find
    • The number of words that have at least one letter repeated.

Approach and Working out
Ways to form words having at least 1 letter repeated = Total number of words that can be formed – words with no letter repeated.
    • = 10 × 10 × 10 × 10 – 10 × 9 × 8 ×7
    • = 10,000 – 5040 = 4960

Hence, option E is the correct answer.

Correct Answer: Option E
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Re: Ten different letters of an alphabet are given, words with 4 letters a [#permalink] New post   18 Jul 2021, 18:11
ScottTargetTestPrep wrote:
Solution:

The total number of 4-letter words that can be formed is 10 x 10 x 10 x 10 = 10,000. The total number of 4-letter words without any repeating letters that can be formed is 10 x 9 x 8 x 7 = 5,040. Therefore, the total number of 4-letter words (with at least one repeating letter) that can be formed is 10,000 - 5,040 = 4,960.

Answer: E

Thanks Scott.
In the last step, I calculated the units digit to arrive at option E.
\(n=10^4-10*9*8*7=10(10^3-9*8*7)\)
Units digit of \(9*8*7=4\)
Now, __0-__4=__6
Hence E.
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Re: Ten different letters of an alphabet are given, words with 4 letters a [#permalink]
18 Jul 2021, 18:11
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