Soupy words

Probability Level pending

If n n is the no of different types of words can you generate from the word SESQUIPEDALIANISM \text{SESQUIPEDALIANISM} , in which :

  • All the vowels occur together
  • Repetition of letters are allowed

Give your answer as sum of all digits of n n

Clarification :

  • If n n is 234797200 234797200 then you would enter 2 + 3 + 4 + 7 + 9 + 7 + 2 + 0 + 0 = 34 2+3+4+7+9+7+2+0+0 = 34 as the answer.
  • SESQUIPEDALIANISM is a real English word meaning having many syllables; given to or characterized by the use of long words


The answer is 18.

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1 solution

Viki Zeta
Oct 17, 2016

SESQUIPEDALIANISM, this word can be rephrased as follows according to the question SSQPDLNSM(EUIEAIAI) So if you consider all the vowels together as 1 letter, you have 10 words. So this can be written in 10! ways. The vowels itself can be written in 8! ways, as there are 8 vowels. Now look at the following SS QPDLN S M( E U I E A I A I) So, S repeats 3 times, E 2 times, I 3 times, A 2 times Total no of ways, n = 10 ! × 8 ! 3 ! × 2 ! × 3 ! × 2 ! n = 1016064000 Sum of digits = 1 + 0 + 1 + 6 + 0 + 6 + 4 + 0 + 0 + 0 = 18 \text{SESQUIPEDALIANISM, this word can be rephrased as follows according to the question} \\ \text{SSQPDLNSM(EUIEAIAI)} \\ \text{So if you consider all the vowels together as 1 letter, you have 10 words.} \\ \text{So this can be written in 10! ways. } \\ \text{The vowels itself can be written in 8! ways, as there are 8 vowels. Now look at the following} \\ \color{#D61F06}{\text{SS}}\color{#333333}{\text{QPDLN}}\color{#D61F06}{\text{S}}\color{#333333}{\text{M}\text{(}}\color{#3D99F6}{\text{E}}\color{#333333}{\text{U}}\color{#20A900}{\text{I}}\color{#3D99F6}{\text{E}}\color{magenta}{\text{A}}\color{#20A900}{\text{I}}\color{magenta}{\text{A}}\color{#20A900}{\text{I}}\text{)} \\ \text{So, S repeats 3 times, E 2 times, I 3 times, A 2 times} \\ \therefore \text{Total no of ways, }n = \dfrac{10! \times 8!}{\color{#D61F06}{3!} \times \color{#3D99F6}{2!} \times \color{#20A900}{3!} \times \color{magenta}{2!}} \\ \implies n = 1016064000 \\ \boxed{\therefore \text{Sum of digits = } 1+0+1+6+0+6+4+0+0+0 = 18}

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