Answer :
Answer:
In combinatorics, combinations refer to the selection of items from a larger set where the order of selection does not matter. The mathematical formula for combinations is used to determine the number of ways to choose a subset of items from a larger set.
Step-by-step explanation:
Combinations Formula
The formula for combinations is given by:
(
,
)
=
!
!
(
−
)
!
C(n,k)=
k!(n−k)!
n!
where:
n is the total number of items in the set.
k is the number of items to be chosen.
!
n! (n factorial) is the product of all positive integers up to
n.
!
k! (k factorial) is the product of all positive integers up to
k.
(
−
)
!
(n−k)! is the factorial of the difference between
n and
k.
Explanation of Each Term
!
n! (n factorial): The number of ways to arrange
n items in a sequence. For example, if
=
5
n=5, then
5
!
=
5
×
4
×
3
×
2
×
1
=
120
5!=5×4×3×2×1=120.
!
k! (k factorial): The number of ways to arrange
k items. This is used to adjust for the fact that the order of the chosen items does not matter in combinations.
(
−
)
!
(n−k)!: The number of ways to arrange the remaining
−
n−k items after the
k items have been chosen. This term accounts for the items that are not selected.
Example of Using the Combinations Formula
Example Problem:
Suppose you have a group of 10 people and you want to select 3 of them to form a committee. How many different ways can you choose 3 people from the 10?
Solution:
Here,
=
10
n=10 (total number of people) and
=
3
k=3 (number of people to choose).
Using the combinations formula:
(
10
,
3
)
=
10
!
3
!
(
10
−
3
)
!
C(10,3)=
3!(10−3)!
10!
Calculate each factorial:
10
!
=
3
,
628
,
800
10!=3,628,800
3
!
=
6
3!=6
(
10
−
3
)
!
=
7
!
=
5
,
040
(10−3)!=7!=5,040
Plug these into the formula:
(
10
,
3
)
=
10
!
3
!
×
7
!
=
3
,
628
,
800
6
×
5
,
040
=
3
,
628
,
800
30
,
240
=
120
C(10,3)=
3!×7!
10!
=
6×5,040
3,628,800
=
30,240
3,628,800
=120
So, there are 120 different ways to choose 3 people from a group of 10.
