Question

The number of ways arranging the letters of the word ‘SUCCESSFUL’ so that all the S’s will come together is:

Answer 1

Hint: Arranging in such a pattern would require the use of permutation in a way where n things are taken all at a time and p of which are alike or one of a kind, q of which are of the other kind, r objects are alike of yet another kind and the remaining objects are different, it can be calculated as:
 $ \dfrac{{\left| \!{\underline {\,
  n \,}} \right. }}{{\left| \!{\underline {\,
  p \,}} \right. \left| \!{\underline {\,
  q \,}} \right. \left| \!{\underline {\,
  r \,}} \right. }} $ .

Complete step-by-step answer:
Therefore we have been given a word successful which has to be arranged so that all the S’s come together, where we observe that the number of S in the word is 3, 2C’s, 2U’s, 1E, 1L and 1F. all together there are 10 letters in the given word where three letters are completely different.
Since we have to arrange so that all the three S’s come together from above we know these 10 letters can be arranged as:
 $
\Rightarrow \dfrac{{\left| \!{\underline {\,
  {10} \,}} \right. }}{{\left| \!{\underline {\,
  3 \,}} \right. .\left| \!{\underline {\,
  2 \,}} \right. .\left| \!{\underline {\,
  2 \,}} \right. }} = \dfrac{{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times \left| \!{\underline {\,
  3 \,}} \right. }}{{\left| \!{\underline {\,
  3 \,}} \right. \times 2 \times 2}} \\
   = \dfrac{{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4}}{4} = 10 \times 9 \times 8 \times 7 \times 6 \times 5 = 151200 \\
 $
Also the three S’s can rearrange themselves in the way like above by:
 $ \dfrac{{\left| \!{\underline {\,
  3 \,}} \right. }}{{\left| \!{\underline {\,
  3 \,}} \right. }} = 1 $
Therefore the required number of ways by which all the S’s in the word ‘SUCCESSFUL’ can be arranged such that they come together are:
 $\Rightarrow \dfrac{{\left| \!{\underline {\,
  {10} \,}} \right. }}{{\left| \!{\underline {\,
  3 \,}} \right. .\left| \!{\underline {\,
  2 \,}} \right. .\left| \!{\underline {\,
  2 \,}} \right. }} \times \dfrac{{\left| \!{\underline {\,
  3 \,}} \right. }}{{\left| \!{\underline {\,
  3 \,}} \right. }} = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 1 = 151200 $
Hence we can conclude that there are 151200 ways to do so.
So, the correct answer is “151200”.

Note: The above rule of finding permutation of arranging p, q alike things from total n things can be extended if in addition to the p or q things there exists s, t and a lot more other things of yet another kind.