How-many-seven-letter-words-can-be-formed-by-using-the-letters-of-the-word-SUCCESS-so-that-neither-two-C-nor-two-S-are-together-

Question Number 21917 by Tinkutara last updated on 06/Oct/17

How many seven letter words can be

formed by using the letters of the word

SUCCESS so that neither two C nor

two S are together ?

Commented by mrW1 last updated on 07/Oct/17

S must be separated by other 4 letters :

YXYXYXYXY

(X means positions for letters UCCE,

Y means possible positions for letter S)

to arrange the 4 letters UCCE in positions

X there are 4! / 2! ways, to place 3 letters

S in positions Y there are C_{3}^{5} ways

\Rightarrow 4! / 2! \times C_{3}^{5}

(2! since letter C repeats twice)

note : in some of these ways the two

letters C are next to each other . to

find the number of ways where two

C are next to each other, we can

proceed as above, but treat CC as a

single letter . that means

YXYXYXY

\Rightarrow 3! \times C_{3}^{4}

\Rightarrow the number of ways such that neither

two S nor two C are next to each other is :

4! / 2! \times C_{3}^{5} - 3! \times C_{3}^{4} = 24 / 2 \times 10 - 6 \times 4 = 96

Commented by Tinkutara last updated on 07/Oct/17

Thank you very much Sir!

Commented by mrW1 last updated on 07/Oct/17

I^{'} m interessed what is the solution in

your book .

Commented by Tinkutara last updated on 07/Oct/17

It does not explain much .