In how many ways can 7 green and 5 yellowballs be arranged in a striaght line when 2 yellow balls are not together?

Explanation

Two yellow balls never occur together, so first we will arrange green balls in a row.
All green balls are similar, so they can arrange in a row in one way.
`xx G xx G xx G xx G xx G xx G xx G xx`
Now we have `7 +1 = 8` places (x) at which we can arrange the yellow balls. In this case, no two yellow balls be together.
`:.` No. of ways to arrange 5 yellow balls at 5 places out of 8 places `= .^(8)C_(5) = 56` and required number of ways `= 1 xx 56 = 56`.