Question

In how many different ways can the letters of the word ‘SALOON’ be arranged,
$i)$ If the two $O$’s must not come together
$ii)$ If the consonants and vowels must occupy alternate places

Answer 1

In the word ‘SALOON’ there are $6$ letters, out of which $4$ letters are distinct and $2$ $O's$ are twice. Now to find out the total number of ways we have to apply the concept of permutation. Finally we get the required answer.

Formula used: ${}^n{p_r} = \dfrac{{n!}}{{(n - r)!}}$

Complete step-by-step solution:
It is given in the question that in the word SALOON , there are $6$ letters out of which $4$ letters are distinct, that is, $S,A,L,N$ and there are $2$ $O$.
Total number of ways $= {}^6{p_4} = \dfrac{{6!}}{{(6 - 4)!}} = \dfrac{{6 \times 5 \times 4 \times 3 \times 2!}}{{2!}} = 360$.
$i)$ Since it is given in the question that the two $O$’s must not come together.
Now we have to consider $2$ $O's$ as one, so now there are $5$ distinct letters.
Therefore the number of ways in which $2$ $O's$ will always be together, Is $5!$ ways $= 5 \times 4 \times 3 \times 2 \times 1 = 120$.
Hence, the number of ways when $2$ $O's$ will never be together,
Total number of ways $-$ Number of ways when they will be together$= 360 - 120 = 240$ ways
$ii)$ Since there are total $6$ letters in the word ‘SALOON’ out of which $3$ are vowels and $3$ are consonants.
As there are $6$ letters, there are 3 odd and $3$ even places and in the question it is stated that the vowels and consonants must occupy the alternate places
So we have to place $3$ places in three places in an alternate manner$3!$ ways$= 6$ ways and $3$ consonants can be placed at three alternate places in $3!$ ways $= 6$ ways.
So, total number of arrangements with vowels and consonants at alternate places $= 6 \times 6 = 36$ ways

Note: In general, permutation can be defined as the act of arranging all the members of a group in an order or sequence.
We can also say that if the group is already arranged, then rearranging of the members is known as the procedure of permuting. It is specifically used where the order of the data matters.
Combination can be defined as the technique of selecting things from a collection in such a manner that the order of selection does not matter.
It is generally used where the order of data does not matter.