Question

How many different permutations can you make with the letters in the word seventeen?

Answer 1

Hint: First we will mention the formula for evaluating the number of permutations if there are $n$ objects with $r$ types. Then evaluate the number of times each term appears. Then finally substitute the values in the formula and evaluate the total number of permutations.

Complete step-by-step answer:
We will mention the formula for the permutation, if there are $n$ objects with $r$ types, then
$\dfrac{{n!}}{{{n_1}!{n_2}!{n_3}!{n_4}!....{n_r}!}}$
The given object is: Seventeen
Here, you can observe that there is a total of $9$ alphabets in the word.
The letter $S$ appears $1$ time.
The letter $E$ appears $4$ time.
The letter $V$ appears $1$ time.
The letter $N$ appears $2$ time.
The letter $T$ appears $1$ time.
We can calculate the different permutations as follows:
 \[
   = \dfrac{{9!}}{{1!4!1!2!1!}} \\
   = \dfrac{{9!}}{{1\times24\times 1\times 2\times 1}} \\
   = \dfrac{{368880}}{{48}} \\
   = 7560 \;
 \]
Hence, the total number of permutations with the letters in the word Seventeen.
So, the correct answer is “7560”.

Note: A permutation is also called an arrangement number or order is a rearrangement of the elements of an ordered list into one-to-one correspondence with itself. A representation of a permutation as a product of permutation cycles is unique up to the ordering of the cycles. Any permutation is also a product of transpositions. Permutations are commonly denoted in lexicographic or transposition order. There is a correspondence between a permutation and a pair of young tableaux known as Schensted correspondence.