Question

4 apples and 3 oranges are arranged in a line. What is the probability that all the oranges are together?

Answer 1

There are total 7 fruits.

It is taken, all fruits are distinguishable.

So, total number of arrangements with them = (7!) = 5040.

Now, when all oranges are together:

There are 5 spaces in and around four apples (3 in between and 2 on sides). For each space, three oranges can be permuted among themselves in (3!) ways = 6 ways and four apples in (4!) ways = 24 ways.

So, probability that all three oranges will be together = {(5 x 6 x 24) / 5040} = (720 / 5040) = (1 / 7) = 0.14.