Question

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 is
\[\begin{align}
  & A.69760 \\
 & B.30240 \\
 & C.99748 \\
 & \text{D}.\text{None of these} \\
\end{align}\]

Answer 1

Hint: For finding the number of words with five letters which have at least one letter repeated out of ten given letters, we will first find the number of total words that can be formed. Then we will find the number of words in which no letter is repeated. Subtracting them, will give us the required number of words. We will use permutation for this. For n arrangements out of which r items are to be arranged, permutation is given by ${}^{n}{{P}_{r}}=\dfrac{n!}{r!}$.

Complete step by step answer:
Here we are given total letters as 10. We have to form words using 5 letters out of the given 10 letters. We need to find the number of words which have at least one letter repeated.
Let us first find the total number of letter words that can be formed using 10 letters. Since repetition is allowed, so all five places can have any of the ten letters. Hence, total words will be equal to $10\times 10\times 10\times 10\times 10={{10}^{5}}$.
Hence total number of words = 100000.
Now let us find the number of words having no letter repeated. Since no letter is repeated and we want to arrange 5 letters out of the given 10 letters, we will use permutation for solving this.
We know, to find number of arrangements of selecting r items out of n items is given as ${}^{n}{{P}_{r}}=\dfrac{n!}{r!}$.
So for arranging 5 letters out of 10 letters, we have,
\[\begin{align}
  & {}^{10}{{P}_{5}}=\dfrac{10!}{5!} \\
 & \Rightarrow \dfrac{10\times 9\times 8\times 7\times 6\times 5!}{5!} \\
 & \Rightarrow 10\times 9\times 8\times 7\times 6 \\
 & \Rightarrow 30240 \\
\end{align}\]
Hence number of words with no repeated letters = 30240.
For finding number of words in which at least one letter is repeated, we need to subtract number of words with no repeated letter from total number of words, we get:
Number of words in which at least one letter is repeated is equal to 100000-30240 = 69760.

So, the correct answer is “Option A”.

Note: Students should not try to find the number of words with one letter repeated, then words with two letters repeated and so on to find the number of letters with at least one letter repeated. Students can get confused between the formula of permutation and combination. ${}^{n}{{P}_{r}}=\dfrac{n!}{r!}\text{ and }{}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$.