krikre wrote:
Could someone pls provide more comprehensive explanation?
Though explanation provided by Chetan is good, I am adding more details to his solution.
Tom has 5 cat-eys (say C) , 5 sulphides (say S) and 3 agates (say A).
So Tom has 5C, 5S and 3A.
The case can fit 5 marbles and please note the question is asking for the arrangement so the order of marbles in Case is also important.
As per the question. We must have 1C, 1S and 1A.
so for remaining 2 spots -
We can Choose 2 of one kind or 1 of two kinds.
1. If we choose 2 of one kind -
We can choose 2 of one kind in 3 ways ( Either 2C or 2A or 2C)
So in total in case, we will have 3 of one kind and 2 other kinds of marble
There is a formula to solve these kinds of arrangement but in case somebody doesn't recall -
We can arrange the 1 kind in 5 ways and second kind by 4. We have 3 identical marble left and 3 spots to fill so the order is not important now.
So ways = 5X4 = 20
so total ways as we can select first 2 marble in 3 ways
= 5X4X3 = 60
2. If we choose 2 kinds of marble to fill 2 empty spaces -
Now we can choose marble in 3 ways ( 1A and 1C, 1A and 1S, 1S and 1C).
So in case, we will 2-2 marbles of 2 kinds and 1 marble of one kind.
Lets' assume we have 2A, 2S and 1C.
for 1C, we can select spots in 5 ways.
for 2As, we can select spots in 6 ways (Select 2 out of 4, order not important).
for remaining 2S, we are left with 2 spots only after fixing 1C and 2As, so only one way.
So ways = 6X5 = 30
so total ways as we can select first 2 marble in 3 ways
= 30X3 = 90.
Total ways = Case 1 + Case 2 = 60 + 90 = 150
Note - It may be easier to solve with the formulas if you are aware of them.