Question

Find the number of ways six '+' and four '-' signs be arranged in a line so that no two '-' signs occur together.

Answer 1

Six ${}^{\prime }{+}^{\prime }$ and four ${}^{\prime }{-}^{\prime }$ are to be arranged. So we will arrange ${}^{\prime }{-}^{\prime }$ in the alternate spaces between ${}^{\prime }{+}^{\prime }$

So we have,

(Refer image)

We have $7$ empty spaces where we have to arrange $4$ ${}^{\prime }{-}^{\prime }$ and so we get,

${}^7{P}_4=7\times 6\times 5\times 4$

But all ${}^{\prime }{-}^{\prime }$ are same so we divide the result by $4!$ and So, we get

$\frac{{}^7{P}_4}{4!}$ possible ways

$=\frac{7\times 6\times 5\times 4}{4\times 3\times 2\times 1}=7\times 5=35$ ways