Question

If all S come together, then in how many ways do the letters of the word SUCCESSFUL be arranged?

Answer 1

Hint: First take the word and find all the letters in it. Now bring all the S’s together. Assume all S’s combined as one letter. Now find the total number of letters present and then find the number of repetitions. Now by using formula number of ways to arrange n things with a, b repetition is,
\[N=\dfrac{n!}{a!b!}\]

Complete step-by-step answer:
Given word in the question is written as given below:
SUCCESSFUL
Number of letters in the above word, can be found to be:
\[\Rightarrow 10\]
The letters which are repeated in the given word are:
\[\Rightarrow \] S, C, U
The number of times the letter S repeated in the word:
\[\Rightarrow \] S’s = 3
The number of times the letter C repeated in the word:
\[\Rightarrow \] C’s = 2
The number of times the letter U repeated in the word:
\[\Rightarrow \] U’s = 2
If we have n things with a, b repetitions we get:
\[\Rightarrow \] \[N=\dfrac{n!}{a!b!}\]
Here we have 10 things with 3, 2, 2 repetitions, we get:
\[\Rightarrow N=\dfrac{10!}{2!3!2!}\]
We will write a list of all letters in word:
\[\Rightarrow \] S, U, C, C, E, S, S, F, U, L
By bringing all S’s together, we get it as follows:
\[\Rightarrow \] S, S, S, U, C, C, E, F, U, L
By combining all the S’s as one letters, we get it as:
\[\Rightarrow \] SSS, U, C, C, E, F, U, L
So, if we arrange the above list of letters all S’s will be together:
Number of letters in the new list above, is given by:
\[\Rightarrow \] 8
The letters which are repeated in the list are:
\[\Rightarrow \] C, U
The number of times C, U are repeated, is written as below:
\[\Rightarrow \] C’s = 2, U’s = 2
We know number of ways to arrange n thing with a, b repetitions:
\[\Rightarrow \] \[N=\dfrac{n!}{a!b!}\]
Here we have 8 things 2, 2 repetitions, we get ways as:
\[\Rightarrow N=\dfrac{8!}{2!2!}\]
By simplifying we get it as follows:
\[\Rightarrow N=\dfrac{40320}{4}=1008\]
Therefore the number of ways is given by 10080.

Note: No need to calculate the first number of ways here we have done just for sake of clarity. After combining, don't consider repetitions in S. Some students think they can arrange inside the list SSS and consider those repetitions but that is completely wrong. So carefully at this step as it is important.