Solution:

To solve this question we can use the concept of combinations and permutations.

Definition:
Combination is a way of selecting items from a collection, such that (unlike permutations) the order of selection does not matter.

Permutation is the arrangement of objects in a specific order.

Step-by-step explanation:

1. Finding the number of ways to select 2 items from a collection of 7:

We can use the formula for combination to find the number of ways to select 2 items from a collection of 7.

nCr = n! / r! * (n-r)!

where n = 7 and r = 2

7C2 = 7! / 2! * (7-2)!

= 21

Therefore, there are 21 ways to select 2 items from a collection of 7.

2. Finding the number of ways to select 2 items from the remaining 5:

After selecting 2 items from a collection of 7, we are left with 5 items.

We can use the same formula to find the number of ways to select 2 items from a collection of 5.

nCr = n! / r! * (n-r)!

where n = 5 and r = 2

5C2 = 5! / 2! * (5-2)!

= 10

Therefore, there are 10 ways to select 2 items from a collection of 5.

3. Finding the number of ways to select 3 items from the remaining 3:

After selecting 2 items from a collection of 7 and 2 items from a collection of 5, we are left with 3 items.

We can use the same formula to find the number of ways to select 3 items from a collection of 3.

nCr = n! / r! * (n-r)!

where n = 3 and r = 3

3C3 = 3! / 3! * (3-3)!

= 1

Therefore, there is only 1 way to select 3 items from a collection of 3.

4. Finding the total number of ways to divide 7 items into 3 groups:

To find the total number of ways to divide 7 items into 3 groups, we need to multiply the number of ways to select 2 items from a collection of 7, the number of ways to select 2 items from a collection of 5, and the number of ways to select 3 items from a collection of 3.

Total number of ways = 21 * 10 * 1

= 210

Therefore, there are 210 ways to divide 7 distinct things into 3 groups, consisting of 2, 2 and 3 things respectively.

3!.2!.2!.3!