show that the number of ways arranging x 1s and y 0s in a li

show that the number of ways arranging x, 1\'s and y, 0\'s in a line such that no two 1\'s are adjacent is C(y + 1 x)

Solution

On a line we have to arrange x,1\'s and y,0\'s.

That means 1\'s and y are always together.

No two 1\'s are adjacent.

x can be arranged in (xC1) ways.

0\'s can be arranged in (1C1) ways.

1\'s and y can be arranged in (yC1) * (1C1) ways.

This can be done by using (yC1) + (1C1) (xC1)

that is C (y + 1 x)