Question

The number of four-letter words can be formed from the letters of the word INFINITY so that 3 are alike, one is different is
 \[\begin{align}
  & \text{(A)}{{\text{ }}^{8}}{{\text{C}}_{4}} \\
 & \text{(B) 4} \\
 & \text{(C) 16} \\
 & \text{(D) 8} \\
\end{align}\]

Answer 1

Hint: From the word INFINITY, we needed to pick four-letter words. This problem can be solved using permutations and combinations concepts. First of all, we should pick a letter which is repeated 3 times from the word INFINITY. We know that the number of ways to select r alike items from n items \[\left( r\le n \right)\] is \[\dfrac{^{n}{{C}_{r}}r!}{r!}\] ways. Now these 3 alike letters should be placed in 4 spaces. We have to find how many of these 3 alike letters can be placed in 4 letters. Thereafter, we should think about the number of ways to choose the fourth letter. We know that by multiplication rule if event A can be done in p ways and event B is done in q ways, then the number of ways that A and B can be done in \[\text{p x q}\] ways. So, by using this multiplication rule, we calculate the number of four-letters formed from the word INFINITY so that 3 are alike and one is different.

Complete step-by-step solution -
In the question, we are given a word INFINITY.
The letters in the word INFINITY are ‘I’, ‘N’, ‘F’, ‘I’, ‘N’, ‘I’, ‘T’, ‘Y’.
In this word INFINITY, we are having 3 ‘I’s, 2 ‘N’s, 1 ‘F’, 1 ‘T’, 1 ‘Y’.
From this word we should find the number of four -letter words in which 3 are alike and one is different.

In the question it was mentioned that 3 letters should be alike, so we should pick a letter from the word INFINITY that should repeat 3 three times. We already know that ‘I’ repeated three times in the word INFINITY. So, we can only select ‘I’ from the word INFINITY.
We needed to form a four-letter word from the INFINITY. So, 3 ‘I’s needed to be placed in 4 ways.
We know that the number of ways to select r alike items from n items \[\left( r\le n \right)\] is \[\dfrac{^{n}{{C}_{r}}r!}{r!}\] ways.
Now these 3 ‘I’s can be placed in 4 places in \[\dfrac{^{4}{{C}_{3}}3!}{3!}\] ways.
Now we should think about in how many ways the fourth letter can be chosen.
In the word INFINITY, we still are having 2 ‘N’s, 1 ‘F’, 1 ‘T’, 1’Y’.
Among these 2 ‘N’s, 1 ‘F’, 1’T’, 1 ‘Y’s, we should select only one letter.
So, we can pick one letter from the letters N, F, T and Y.
The number of letters to select the fourth letter is 4 ways.
According to multiplication rule, if event A and events B occurred in p and q ways respectively in a simultaneous manner. The total number of ways of occurrence of both p and q simultaneously is equal to \[\text{p x q}\].
Hence, the total number of ways we can form four-letter words from the word INFINITY in which three words are alike and one is different = (4)(4) =16 ways.

Note: Students must be careful using the addition rule (or) multiplication rule during solving the question based on permutations and combinations concept.
The multiplication rule states that if event A and events B occurred in p and q ways respectively. Then, the total number of ways of occurrence of both p and q simultaneously is equal to \[\text{p x q}\].
The addition rule states that if event A and events B occurred in p and q ways respectively. Then, the total number of ways of occurrence of either p or q is equal to \[\text{p + q}\].