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Ten different letters of an 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 are (1982 - 2 Marks)
- a)69760
- b)30240
- c)99748
- d)none of these
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Ten different letters of an alphabet are given. Words with five letter...
Total number of words that can be formed using 5 letters out of 10 given different letters
= 10 × 10 × 10 × 10 × 10 (as letters can repeat)
= 1, 00, 000
Number of words that can be formed using 5 different letters out of 10 different letters
= 10P5 (none can repeat)
= 10 × 10 × 10 × 10 × 10 (as letters can repeat)
= 1, 00, 000
Number of words that can be formed using 5 different letters out of 10 different letters
= 10P5 (none can repeat)
∴ Number of words in which at least one letter is repeated
= total words–words with none of the letters repeated
= 1,00,000 – 30,240
= 69760
= total words–words with none of the letters repeated
= 1,00,000 – 30,240
= 69760
Most Upvoted Answer
Ten different letters of an alphabet are given. Words with five letter...
Problem:
We are given ten different letters of an alphabet. We need to form words with five letters using these given letters. We need to find the number of words that have at least one letter repeated.
Solution:
To find the number of words that have at least one letter repeated, we can use the principle of inclusion-exclusion.
Principle of Inclusion-Exclusion:
The principle of inclusion-exclusion is a counting technique used to determine the size of a set that is a union of several other sets.
In this problem, we will use the principle of inclusion-exclusion to count the number of words with at least one letter repeated.
Step 1: Total Number of Words:
To find the total number of words that can be formed using the given letters, we need to select 5 letters out of the 10 given letters. This can be done in C(10, 5) ways.
C(10, 5) = 10! / (5! * (10-5)!) = 10! / (5! * 5!) = (10 * 9 * 8 * 7 * 6) / (5 * 4 * 3 * 2 * 1) = 252
So, the total number of words that can be formed using the given letters is 252.
Step 2: Number of Words with No Repeated Letters:
To find the number of words with no repeated letters, we need to select 5 distinct letters out of the 10 given letters. This can be done in C(10, 5) ways, which we have already calculated to be 252.
So, the number of words with no repeated letters is 252.
Step 3: Number of Words with At Least One Letter Repeated:
To find the number of words with at least one letter repeated, we need to subtract the number of words with no repeated letters from the total number of words.
Number of words with at least one letter repeated = Total number of words - Number of words with no repeated letters
Number of words with at least one letter repeated = 252 - 252 = 0
So, the number of words that have at least one letter repeated is 0.
Therefore, the correct answer is option A) 0.
We are given ten different letters of an alphabet. We need to form words with five letters using these given letters. We need to find the number of words that have at least one letter repeated.
Solution:
To find the number of words that have at least one letter repeated, we can use the principle of inclusion-exclusion.
Principle of Inclusion-Exclusion:
The principle of inclusion-exclusion is a counting technique used to determine the size of a set that is a union of several other sets.
In this problem, we will use the principle of inclusion-exclusion to count the number of words with at least one letter repeated.
Step 1: Total Number of Words:
To find the total number of words that can be formed using the given letters, we need to select 5 letters out of the 10 given letters. This can be done in C(10, 5) ways.
C(10, 5) = 10! / (5! * (10-5)!) = 10! / (5! * 5!) = (10 * 9 * 8 * 7 * 6) / (5 * 4 * 3 * 2 * 1) = 252
So, the total number of words that can be formed using the given letters is 252.
Step 2: Number of Words with No Repeated Letters:
To find the number of words with no repeated letters, we need to select 5 distinct letters out of the 10 given letters. This can be done in C(10, 5) ways, which we have already calculated to be 252.
So, the number of words with no repeated letters is 252.
Step 3: Number of Words with At Least One Letter Repeated:
To find the number of words with at least one letter repeated, we need to subtract the number of words with no repeated letters from the total number of words.
Number of words with at least one letter repeated = Total number of words - Number of words with no repeated letters
Number of words with at least one letter repeated = 252 - 252 = 0
So, the number of words that have at least one letter repeated is 0.
Therefore, the correct answer is option A) 0.
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Ten different letters of an 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 are (1982 - 2 Marks)a)69760b)30240c)99748d)none of theseCorrect answer is option 'A'. Can you explain this answer?
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