Question

A number of four-letter words that can be formed with the letters in the word EQUATION with at least one letter repeated are:
A. $2400$
B. $2408$
C. $2416$
D. $2432$

Answer 1

In the given question first we have to count the total number of alphabets is the given word EQUATION. Then we have to find the number of four-letter words that can be formed with repetition, Number of four-letter words that can be formed without repetition. Then we can calculate the number of four-letter words that can be formed at least with one repetition. This we get the correct answer.

Complete step by step Answer:

From the given question we can write that:
Number of distinct alphabets in the word EQUATION is $8$
Now we have to make a number of four letter words
Number of four letter words that can be formed with repetition$= 8 \times 8 \times 8 \times 8 = 4096$
Because the total number of letters is eight.
Now again according to the question:
Number of four letter words that can be formed without repetition$= 8 \times 7 \times 6 \times 5 = 1680$
Now:
Number of four-letter words that can be formed at least with one repetition$=$ (Number of four-letter words that can be formed with repetition)-(Number of four-letter words that can be formed without repetition)
$\Rightarrow 4096 - 1680 = 2416$
Thus the number of four-letter words that can be formed with the letters in the word EQUATION with at least one letter repeated is $2416$.
Thus we get the correct answer, Hence the correct option is C.

Note: In the given question we have to remember the formula for word can be formed with one repetition because this formula is used in the given problem i.e. Number of four-letter words that can be formed at least with one repetition$=$ (Number of four-letter words that can be formed with repetition)-(Number of four-letter words that can be formed without repetition). We have to apply this formula accordingly in a given question and we get the correct answer.