Question

10 different letters of an alphabet are given. Words with 5 letters are formed from these given letters, then the number of words which have at least one letter repeated is
(a) 69760
(b) 30240
(c) 99748
(d) none

Answer 1

Hint: In this question, we need to find the total number of 5 letter words that are possible from 10 letters using the formula of number of functions \[{{n}^{m}}\]and also find the 5 letter words in which all the letters are different using the permutations formula \[{}^{n}{{P}_{r}}\]. Now, we have to subtract those two values to get the 5 letter words in which at least one letter is repeated.

Complete step-by-step answer:
Now, from the given conditions in the question we have 10 letters out of which we need to form 5 letter words
Now, let us find the total number of 5 letter words possible using 10 different letters
As we already know that the total number of functions from A to B is given by the formula when n is the number elements in A and m is the number elements in B
\[{{n}^{m}}\]
Now, the number of 5 letter words using 10 letters are
Now, on comparing with the above formula we get,
\[n=10,m=5\]
Now, on substituting the respective values in the above formula we get,
\[\Rightarrow {{10}^{5}}\]
Now, let us find the number of 5 letter words possible with all letters different
As we already know that this can be found by the permutation formula as it is the arrangement of 5 letters using 10 letters which is given by the formula
\[{}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}\]
Now, we need to arrange 5 letters words using 10 letters which is done using the above formula.
Now, on comparing the given conditions with the formula we get,
\[n=10,r=5\]
Now, on substituting these values in the respective formula we get,
\[\Rightarrow {}^{10}{{P}_{5}}\]
Now, this can be further written as
\[\Rightarrow \dfrac{10!}{\left( 10-5 \right)!}\]
Now, on further simplification we get,
\[\Rightarrow 10\times 9\times 8\times 7\times 6\]
Let us now find the number of 5 letter words with at least one letter repeated
Total number of 5 letter words with at least 1 letter repeated is the subtraction of total number of 5 letter words possible and number of 5 letter words with all different letters
Now, on substituting the respective values in the above condition we get,
\[\Rightarrow {{10}^{5}}-{}^{10}{{P}_{5}}\]
Now, this can be further written as
\[\Rightarrow {{10}^{5}}-10\times 9\times 8\times 7\times 6\]
Now, this can be further written in the simplified form as
\[\Rightarrow 100000-30240\]
Now, on further simplification we get,
\[\Rightarrow 69760\]
Hence, the correct option is (a).


Note:
Instead of subtracting the 5 letter words with no letter repeated from the total number of 5 letter words possible we can also find it by finding the 5 letters words with 1letter repeated, 2 letters repeated, 3 letters repeated and so on and then add all of them. Both the methods give the same result.
It is important to note that the number of 5 letter words with no repeated is the arrangement of 5 letters out of 10 letters in different ways but not the combinations so we use a permutation formula.