Ten different letters of alphabet are given, words with $5$ letters are formed from these given letters. Then, what is the number of words which have at least one letter repeated?

Well I do understand some part of the question.
The only confusing thing is that if we had chosen different letters of alphabet, how can we form words which have the letters repeated.

For the record, I am new to permutation and combination, so please put the explanation to the most basic level.


Let's say you have been given ten letters A,B,C,D,E,F,G,H,I,J.

Some of the 5-lettered words which could be formed using these 10 letters are ABCDE, GEFIB, etc. However, words like AABCE, BEEIJ, or even AAAAA, are acceptable words.

The question demands you to find out the number of words which have at least one letter repeated. We have to work it out in a reverse way.

First, let's find out what is the total number of 5-lettered words that can be formed from the given 10 letters without any condition.

Consider 5 blank spaces.$$-----$$

There are 10 ways to fill in the first blank, that is, we can put any letter of the given 10 in the first blank. Similarly, we also have 10 ways to fill the second blank space, since we have put no condition regarding the arrangement of letters in these spaces. Same goes for the third, fourth, and the fifth blank spaces. By the Fundamental Theorem of Multiplication, we have the total number of words so formed as $10*10*10*10*10=10^5$

Now, let's put a condition that when we fill in these blank spaces, we must assure that no letter is repeated. Doing so, we realise that we still have 10 options for the first blank space. However, for the second blank, we have only 9 options, since we have already used one of the letters in the first blank and we are not allowed to use it again. Continuing further, we have 8 options for the third blank, 7 for the fourth, and 6 for the fifth. Again, by the Fundamental Theorem of Multiplication, we have the total number of words as $10*9*8*7*6=30240$

So far, we have found the total number of words that can be made using the given 10 letters, and the number of words that are formed when no letter is repeated. The remaining possibilities consist of words which have at least one letter is repeated. So, all you have to do is subtract the second result from the first one, and you have your answer.

If you're familiar with the use of $\;^nP_r$, and $\;^nC_r$, use them to get answer faster. :)


Hint: The number of $5$-letter words where the letters are selected from a $10$-letter alphabet (possibly with repetition) is $10^5$.

The number of $5$-letter words with all letters distinct is $(10)(9)(8)(7)(6)$.


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