In-how-many-ways-can-6-players-be-lined-up-if-2-particlar-players-must-not-stand-next-to-each-other-

Question Number 153572 by otchereabdullai@gmail.com last updated on 10/Sep/21

In how many ways can 6 players be

lined up if 2 particlar players must

not stand next to each other

Commented by Tawa11 last updated on 08/Sep/21

Let the players be ABCDEF

\Rightarrow * C * D * E * F * [A and B are not next to each other]

∴ = 4! \times 5 \times 4

∴ = 24 \times 20

∴ = 480 ways .

Answered by mr W last updated on 08/Sep/21

X A Y B X

X = z e r o o r m o r e p l a y e r s

Y = o n e o r m o r e p l a y e r s

2 X + Y = 4

{(1 + x + x^{2} + \dots)}^{2} (x + x^{2} + x^{3} + \dots)

= \frac{x}{{(1 - x)}^{3}} = \underset{k = 0}{\sum^{\infty}} C_{2}^{k + 2} x^{k + 1}

c o e f . o f x^{4} i s C_{2}^{3 + 2} = C_{2}^{5} = 10 .

t o t a l l y w a y s :

C_{2}^{5} \times 2! \times 4! = 480

Commented by mr W last updated on 08/Sep/21

C_{2}^{5} = 10 :

A P B P P P

A P P P B P

A P P B P P

A P P P P B

P A P B P P

P A P P B P

P A P P P B

P P A P B P

P P A P P B

P P P A P B

y o u c a n a l s o l i n e u p t h e o t h e r 4 p l a y e r s,

_P_P_P_P_

a n d t h e n p l a c e t h e t w o p a r t i c u l a r

p l a y e r s a t t h e f i v e_p o s i t i o n s, l i k e

_P A P_P_P B . t h e r e a r e C_{2}^{5}

p o s s i b i l i t i e s .

Commented by otchereabdullai@gmail.com last updated on 08/Sep/21

may God continue to bless you always

prof W

Answered by peter frank last updated on 08/Sep/21

thank you both