Question

How many ways can the word “Math” be arranged?

Answer 1

Hint: To solve this question we will use the concept of permutation and combinations. First we will count the letters in the word math. Then we will use the permutation formula with repeating letters. The permutation formula is given by ${}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$

Complete step by step solution:
We have been given a word math.
We have to find the number of ways the word math can be arranged.
Now, we know that a permutation is the act of arranging the objects or numbers in order.
Now, let us analyze the word math, it has 4 letters and no letter is repeated.
So we will simply count the letters and solve further.
We know that to arrange 4 letters we have $4!$ ways.
Now, we know that we can solve factorial as $n!=(n-1)!(n-2)!......2\times 1$
Now, solving the $4!$ we will get
\[\begin{align}
  & \Rightarrow 4\times 3\times 2\times 1 \\
 & \Rightarrow 24 \\
\end{align}\]
Hence there are 24 ways the word math can be arranged.

Note:
If there are repeated letters then we have to divide the total factorial by number of repetitions. Also there is a difference between permutation and combinations. Generally permutation can be defined as the act of arranging objects in an order. Whereas combination is the technique of selecting objects from the collection of objects such as the order of selection does not matter. The order of selection does not matter in permutation.