If-n-objects-are-arranged-in-a-row-then-find-the-number-of-ways-of-selecting-three-of-these-objects-so-that-no-two-of-them-are-next-to-each-other-

Question Number 21587 by Tinkutara last updated on 28/Sep/17

If n objects are arranged in a row, then

find the number of ways of selecting

three of these objects so that no two of

them are next to each other .

Commented by mrW1 last updated on 29/Sep/17

Now I understand the question .

Answered by mrW1 last updated on 06/Oct/17

to select 3 objects from n objects there

are totally C_{3}^{n} = \frac{n (n - 1) (n - 2)}{6} ways

for 3 objects next to each other there

are (n - 2) ways

for 2 objects next to each other there

are 2 (n - 3) + (n - 3) (n - 4) = (n - 2) (n - 3) ways

\Rightarrow number of ways without two objects

next to each other is therefore

\frac{n (n - 1) (n - 2)}{6} - (n - 2) - (n - 2) (n - 3)

= \frac{n (n - 1) (n - 2)}{6} - {(n - 2)}^{2}

= \frac{(n - 2) (n^{2} - 7 n + 12)}{6}

= \frac{(n - 2) (n - 3) (n - 4)}{6}

= C_{3}^{n - 2}

Commented by Tinkutara last updated on 06/Oct/17

Thank you very much Sir!