Question

How many meaningful English words can be formed with the letters ESRO using each letter only once in each word?
A). None
B). One
C). Two
D). Three

Answer 1

Hint: First we’ll try to figure out the concept of the question. Whenever we have to do arrangements, we’ll always try to use permutations. The reason being if we have to arrange ‘r’ items from n items then we always use a formula of permutation.

Complete step-by-step solution:
In this question, we have given ‘a’ word ESRO. It has 4 letters namely E, S, R, O. If we have to just rearrange letters then we can use permutation formula to get this, ${}^n{P_r} = \dfrac{{n!}}{{(n - r)!}}$. Since we are taking four out of four so ‘n’ and ‘r’ both will be equal to 4.
${}^4{P_4} = \dfrac{{4!}}{{(4 - 4)!}} = \dfrac{{4!}}{{0!}} = 4! = 24$
But, it’s not the correct answer. Because there will be many arrangements which will not make any sense. Hence we need to arrange the terms manually.
Terms which will have meaning are ROSE- it’s a flower, SORE-painful, EROS- it’s one of the four ancient Greek terms which means love.
Hence, there are only three possible arrangements of letter E, R, S, O which have some meaning. So, Option (d) is the correct answer.

Note: Many students get confused when to use permutation and when to use the combination. A combination is always used to choose the items. For example, if we have n items and we have to choose ‘r’ out of that then we’ll use the combination. Its formula is ${}^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$. On the other hand, the permutation is used for arrangements.