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 !
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 !
