Answer:
Explanation:
Step 1: Concept Introduction
Sum rule: If an event can occur either in ways or in ways (non-overlapping), the number of ways the event can occur is then m+n.
The definition of permutation (order is important) is –
No repetition allowed:
Repetition allowed:
The definition of combination (order is important) is –
No repetition allowed:
Repetition allowed:
With
.
Distributing distinguishable objects into k distinguishable boxes such that
; objects are placed in the box
can be done in
ways.
Step 2: Non-integer solution
(a)
The integer solutions of the equation
can be obtained by selecting r objects from a set with n objects such that there are
chosen from the first type,
are chosen from the second type and so on.
Thus, the number of solutions can then be obtained by using the definition of a combination (since the order of the solutions is not important) and repetition is allowed (since more than one
value can take on the same value) –
Therefore, the result is obtained as
.
Step 3: Number of non-negative integer solutions
(b)
It is given that –
It is needed to select 17 indistinguishable objects from 4 distinguishable boxes (variables).
Here n=4, r=17 .
Since repetition is allowed, so substitute the value and calculate –
Therefore, the result is obtained as 1140.
Step 4: Number of positive integer solutions
(c)
It is given that –
All variables have to be at least 1. Then redefine
and
(which are then 4 variables that are at least 0 ).
It is needed to select 13 indistinguishable objects from 4 distinguishable boxes (variables).
Here n=4, r=3 .
Since repetition is allowed, so substitute the value and calculate the –
Therefore, the result is obtained as 560.