Question

Eight different letters of the alphabet are given. Words of four letters from these are formed. The number of such words with at least one letter repeated is:
1. \[(\dfrac{8}{4})-{}^{8}{{P}_{4}}\]
2. \[{{8}^{4}}+(\dfrac{8}{4})\]
3. \[{{8}^{4}}-{}^{8}{{P}_{4}}\]
4. \[{{8}^{4}}-(\dfrac{8}{4})\]

Answer 1

Hint: To solve this question firstly decide whether to use the permutation or combination. But in the given question we have to arrange the alphabets so use the concept of permutation. To find at least one repetition use the approach total numbers of letters when repetition is allowed subtract number of letters when no repetition is allowed.

Complete step by step answer:
The given question is from permutation and combination. Now we have to decide whether to use permutation or combination for solving this question.
Let us get the knowledge of both the terms then we are able to decide which concept we should use.
Permutation: It is the method of arranging all the elements given into some order or some kind of sequence. To arrange the elements there are two ways i.e. when repetition of the elements are allowed and another one is when repetition of the elements are not allowed.
Formula for permutation is: \[^{n}{{P}_{r}}=\dfrac{n!}{(n-r)!}\]
Where,
\[n\]is the total number of elements.
\[r\]is the number of elements we want to arrange.
Combination: It is the method of selecting the items from the given collection. In this the order or sequence does not affect the final answer.
Formula used for the combination is: \[^{n}{{C}_{r}}=\dfrac{n!}{r!(n-r)!}\]
Where,
\[n\]is the total number of elements.
\[r\]is the number of elements we want to select.
After knowing both the terms, it is clear that in the given question we have arrange the eight alphabets so we will use permutation to solve this,
We have to find the letter where at least one alphabet is repeated. To find this firstly know the meaning of the word at least. This simply means that not less than i.e. always equal to or greater than the given value.
So if we are able to find a total number of four letter words from eight alphabets and letters with at least one repetition not allowed. Then by subtracting both the terms we can get your final answer.
Total number of four letter word from eight alphabets \[=8\times 8\times 8\times 8\]
\[\Rightarrow {{8}^{4}}\]
Total number of letters when no repetition is allowed \[=8\times 7\times 6\times 5\]
\[{{\Rightarrow }^{8}}{{P}_{4}}\]
So, total number of four letter words with at least one letter repeated \[={{8}^{4}}{{-}^{8}}{{P}_{4}}\]

So, the correct answer is “Option 3”.

Note: We must know the difference between the terms at least and at most. Both the terms are a little bit confusing, you must have a clear picture of both the terms because their meanings are very different. At least means not less than whereas at most means not more than. Deal carefully with both the terms.