The problem states:

We are given 5 identical blue marbles, 4 identical green marbles, and 2 identical red marbles. The condition is that the green marbles must be placed together. We are asked to find how many different ways these marbles can be arranged side by side under this condition.

Solution approach:

1. Treat the block of green marbles as a single unit: Since the green marbles must be together, we can treat them as a single "super marble". This simplifies the total number of objects to arrange.

   Now, instead of arranging 5 blue, 4 green, and 2 red marbles, we are arranging:

   • 5 blue marbles,
   • 1 "green block",
   • 2 red marbles.

   In total, we now have $5 + 1 + 2 = 8$ objects.

2. Number of arrangements: The number of ways to arrange 8 objects where some are identical is given by the formula for permutations of multiset:

   $\text{Total arrangements} = \frac{8!}{5! \times 2! \times 1!}$

   where:

   • $8!$ is the factorial of the total number of objects,
   • $5!$ accounts for the identical blue marbles,
   • $2!$ accounts for the identical red marbles,
   • $1!$ accounts for the green block.

Let's calculate this.The total number of different ways to arrange the marbles, with the condition that the green marbles are together, is 168.

Would you like more details or have any questions?

Here are 5 related questions for further practice:

1. What if the green marbles did not need to be together? How many arrangements would there be?
2. How would the number of arrangements change if there were 3 red marbles instead of 2?
3. What is the number of ways to arrange 4 identical green marbles and 4 identical blue marbles with no additional condition?
4. How many ways can we arrange 6 identical blue marbles and 3 identical green marbles if the green marbles must be separated?
5. How many ways can we arrange 5 blue marbles, 3 green marbles, and 2 red marbles if there is no restriction on their positions?

Tip: When dealing with identical objects in permutation problems, always remember to divide by the factorial of the number of identical items.