Question

In how many ways can 9 different colour balls be arranged in a row so that black, white, red and green balls are never together

Answer 1

Total number of ways in which 9 different colour balls can be arranged in a row
=9!  ⋯=9!  ⋯(A)

Now we will find out total number of ways in which 9 different colour balls can be arranged in a row so that black, white, red and green balls are always together.

We have total 9 balls. Since black, white, red and green balls are always together, group these 4 balls together and consider as a single ball. Hence we can take total number of balls as 6. These 6 balls can be arranged in 6! ways.

We had grouped 4 balls together. These 4 balls can be arranged among themselves in 4!ways.

Hence, total number of ways in which 9 different colour balls be arranged in a row so that black, white, red and green balls are always together
=6!×4!  ⋯=6!×4!  ⋯(B)

From (A) and (B),
Total number of ways in which 9 different colour balls can be arranged in a row so that black, white, red and green balls are never together
=9!–6!×4!
=6!×7×8×9−6!×4!
=6!(7×8×9–4!)
=6!(504–24)
=6!×480
=720×480
=345600

Answer 2

Concept Introduction:

A probability analysis is a branch of mathematics that deals with numerical descriptions of how likely it is that a particular event will take place, or whether a certain proposition will be true.

Given: 9 different colour balls be arranged in a row so that black, white, red and green balls are never together.

To Find:

In how many ways can 9 different colour balls be arranged in a row so that black, white, red and green balls are never together

Solution:

According to problem:

=9!–6!×4!

=6!×7×8×9−6!×4!

=6!(7×8×9–4!)

=6!(504–24)

=6!×480

=720×480

=345600

Final Answer:

There are 345600 ways.

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