Cryto Eth

How many ways can the letters of the word square be arranged if all the vowels are together?

In how many different ways, can the letters of the words EXTRA be arranged so that the vowels are never together?

Answer [Detailed Solution Below]

Option 4 : 72

Calculation:

EXTRA → Total number of words = 5 and total number of vowels = 2

The word EXTRA can be arranged in 5! ways = 120 ways

The word EXTRA can be arranged in such a way that the vowels will be together = 4! × 2!

⇒ [4 × 3 × 2 × 1] × [2 × 1]

⇒ 48 ways

The letters of the words EXTRA be arranged so that the vowels are never together = [120 - 48] = 72 ways.

∴ The letters of the words EXTRA be arranged so that the vowels are never together in 72 ways.

Answer

Nội dung chính

Permutations of repeated letters in a word
How many ways a letter can be arranged?
How many ways can the letters of the word permutation can be arranged?
How many ways the word over expand can be arranged?
How many ways can the letters of the word square be arranged if all the vowels are together?

Verified

Hint: We have permutations and combinations here. it means we are going to rearrange letters by taking one letter or all letters. But we have to be careful as there are many repeated letters. So there will be repeated words as well. Generally when there n different letters in a word, the total number of ways in which all the letters are rearranged is n! ways.

Complete step-by-step solution:
In the word MISSISSIPPI, there are 4 I’s, 2 P’s, 4 S’s.
And the total number of letters including the repetitions is 11 letters.
So the total number of ways in which it can arrange is 11!. But we have to account for all the repeated letters.
So now we have to divide it by 4!, 2!, 4!.
We have to pretend that there are no repeated letters. And then just basically divide it by 4!, 2!, 4! To eliminate all the words with repeated letters
There is a formula for this that is available in permutations and combinations. It states the following :
The number of permutation of n things taken all at a time, in which p things are alike of one kind q is alike of the second kind and r things are alike of the third kind and rest are different is :
$\Rightarrow \dfrac{n!}{p!q!r!}$
So we get the number of ways to arrange MISSISSIPPI is $\dfrac{11!}{4!2!4!}$.
And on further simplification, we get
$\Rightarrow \dfrac{11\times 10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{\left[ 4\times 3\times 2\times 1 \right]\left[ 2\times 1 \right]\left[ 4\times 3\times 2\times 1 \right]}$
Now we know that the same terms from numerator and denominator cancels out. Therefore, we get
$\Rightarrow 11\times 10\times 9\times 7\times 5$
$\Rightarrow 34650$
$\therefore $ Hence the number of ways can the letters in ‘MISSISSIPPI’ be arranged is 34650.

Note: Permutations and Combinations is a very tricky chapter and one needs a lot of practice for it. There is a lot of logic and understanding required to solve this kind of question in the limited amount of time. Though there are formulae, it is important to understand the logic behind the question rather than memorizing the formulae. In this way, though we may have trouble solving the first few questions , the rest would gradually become simple.

Donate

GCSEA-Level

Tutorial Contents / Maths / Permutations of repeated letters in a word

Permutations of repeated letters in a word

In this video tutorial I show you how to calculate how many arrangements or permutations there are of letters in a word where a letter is repeated.
The following examples are given.
[1] In how many ways can the letters in the word EYE be arranged?
[2] In how many ways can the letters in the word STATISTIC be arranged?

MichaelExamSolutionsKid2020-02-28T11:44:08+00:00

How many ways a letter can be arranged?

=360 the number of ways.

How many ways can the letters of the word permutation can be arranged?

Thus total number of permutations = 14×1814400=25401600.

How many ways the word over expand can be arranged?

Required number of ways = [120 x 6] = 720.