P & C concept 1

Box - Ball concept 

In how many ways one can arrange

(a) 5 different balls in 3 different  boxes    --- very common in cat
(b) 5 identical balls in 3 different  boxes    -- very common in cat
(c) 5 identical balls  in 3 identical boxes    --  less likely to appear in cat
(d) 5 different balls  in 3 identical boxes    --   might appear in cat 

(a) 5 different balls in 3 different  boxes   : (This is easy one)

Each ball can go to one of the 3 boxes. So, it has 3 choices. So, it will be 3*3*3*3*3 choices for each.
So, 3^5

Thumb rule : where^what 

5 person in lift of 8 floors . 

where --> lift , what --> person --> 8^5

(b) 5 identical balls in 3 different  boxes :

It is like this:
a+b+c+d = 4
where a, b, c and d are number of balls in each  boxes .

So, 7C3 ways


Thumb rule : counting number of whole number solutions .. 

a+b+c = n --> C(n +r -1 , r-1)

(c) 5 identical balls in 3 identical  boxes    : 

5 ways: (5, 0, 0), (3, 2, 0), (4, 1, 0),(2,2,1) or (1, 1, 3)

Thumb rule : Just counting number of ways manually 

(d) 5 different balls in 3  identical  boxes   (This is calculation based..arguably toughest one of the four )

(5, 0, 0) => 1 way only 

(4, 1, 0) =>  5C4 ways
4 balls to be grouped can be chosen in 5C4 ways and remaining one in 1 way

(2, 3, 0) => 2 balls can be chosen in 5C2 ways.

(2, 2, 1) => 5C2*3c2 /2 ways 
5C2 ways to chose 2 balls and then select 2 balls in 3 in 3c2 ways .. divide by 2 for repetitive pairs .. 

(1, 1, 3) => 5C3 ways 

In total 10 + 5 + 10 + 15 + 1 = 41 ways

Thumb rule : Just counting number of ways manually and then arrange them
Please leave a comment in case of queries .. 

Cheers and good luck !