A combination is a selection of all or part of a set of objects, without regard to the order in which objects are selected. For example, suppose we have a set of three letters: A, B, and C. we might ask how many ways we can select 2 letters from that set.
Combination is defined and given by the following function:
C(n,r)=n!r!(n−r)!C(n,r)=n!r!(n−r)!
Where −
· nn = the number of objects to choose from.
· rr = the number of objects selected.
Problem Statement:
How many different groups of 10 students can a teacher select from her classroom of 15 students?
Solution:
Step 1: Determine whether the question pertains to permutations or combinations. Since changing the order of the selected students would not create a new group, this is a combinations problem.
Step 2: Determine n and r
n = 15 since the teacher is choosing from 15 students.
r = 10 since the teacher is selecting 10 students.
Step 3: Apply the formula
15C10=15!(15−10)!10!=15!5!10!=15(14)(13)(12)(11)(10!)5!10!=15(14)(13)(12)(11)5!=15(14)(13)(12)(11)5(4)(3)(2)(1)=(14)(13)(3)(11)(2)(1)=(7)(13)(3)(11)=3003
Each of several possible ways in which a set or number of things can be ordered or arranged is called permutation Combination with replacement in probability is selecting an object from an unordered list multiple times.
Combination with replacement is defined and given by the following probability function:
nCr=(n+r−1)!r!(n−1)!nCr=(n+r−1)!r!(n−1)!
Where −
· nn = number of items which can be selected.
· rr = number of items which are selected.
· nCrnCr = Unordered list of items or combinations
Problem Statement:
There are five kinds of frozen yogurt: banana, chocolate, lemon, strawberry and vanilla. You can have three scoops. What number of varieties will there be?
Solution:
Here n = 5 and r = 3. Substitute the values in formula,
nCr=(n+r−1)!r!(n−1)! =(5+3+1)!3!(5−1)! =7!3!4! =50406×24 =35