 # In how many ways can the letters of the word ‘CHEESE’ be arranged?

Question:

In how many ways can the letters of the word ‘CHEESE’ be arranged?

Solution:

Given: We have 6 letters

To Find: Number of words formed with Letter of the word ‘CHEESE.’

The formula used: The number of permutations of $n$ objects, where $p_{1}$ objects are of one kind, $p_{2}$ are of the second kind, ..., $p_{k}$ is of a $k^{\text {th }}$ kind and the rest if any, are of a

different kind is $=\frac{n !}{p_{1} ! p_{2} ! \ldots \ldots \ldots \ldots \ldots p_{k} !}$

Suppose we have these words $-\mathrm{C}, \mathrm{H}, \mathrm{E}_{1}, \mathrm{E}_{2}, \mathrm{~S}, \mathrm{E}_{3}$

Now if someone makes two words as $\mathrm{CHE}_{1} \mathrm{E}_{3} \mathrm{SE}_{2}$ and $\mathrm{CHE}_{2} \mathrm{E}_{3} \mathrm{SE}_{1}$

These two words are different because $E_{1}$. $E 2$ and $E_{3}$ are different but we have three

similar E's hence, in our case these arrangements will be a repetition of same words.

In the word CHEESE, 3 E's are similar

$\therefore \mathrm{n}=6, \mathrm{p}_{1}=3$

$\Rightarrow \frac{6 !}{3 !}=\frac{720}{6}=120$

In 120 ways the letters of the word 'CHEESE' can be arranged.