Question

How many permutations of the letters of the word APPLE are there?
A. 40
B. 50
C. 60
D. 70

Answer 1

Hint: Here, total number of letters is 5 (i.e. $A, P, P, L, E$). In this word, the letter $P$ is two times. We can find the number of permutations using the permutation formula. But also keep in mind that letter $P$ is two times. Simplify the factorial to get the answer.

Complete step by step answer:
There are 5 letters in the word $\text{APPLE}$ out of which 2 are $P$’s and the rest are all distinct letters.
Here, we can use formula, \[\dfrac{{n!}}{{p!}}\]
Where $n$ is the total number of letters, and $p$ is the number of times a particular letter is repeated.
Permutations of the letters of the word APPLE =\[\dfrac{{5!}}{{2!}} = \dfrac{{5 \times 4 \times 3 \times 2 \times 1}}{{2 \times 1}} = 60\].
[Since, $5! = 5 \times 4 \times 3 \times 2 \times 1$ and $2! = 2 \times 1$]

Therefore, the permutations of the letters of the word APPLE are 60. So, the correct option is (C).

Note:
In these types of questions, first check whether a question is asked about combination or permutation. Permutation means arrangement of things, and combination means taking a particular number of items at a time (arrangement does not matter in combination). Then apply the proper formula as required. Permutation means arrangement of letters. But be careful about repeated letters. If one, two or and any numbers of letters repeated, use the formula for the same. Also, if you get a factorial of the large number, never find the value of that factorial. Cancel the two factorials using proper rules. If in any case you get the final result in factorial of the large number, then leave the result in factorial form, don’t find the value.