Find the value of the combination 10C2
The Correct Answer and Explanation is:
To find the value of ( 10C2 ), which represents the number of ways to choose 2 items from a set of 10 items without regard to the order of selection, we use the combination formula:
[
nCr = \frac{n!}{r!(n – r)!}
]
In this formula:
- ( n ) is the total number of items to choose from (in this case, ( n = 10 )).
- ( r ) is the number of items to choose (in this case, ( r = 2 )).
- ( n! ) (n factorial) is the product of all positive integers up to ( n ).
Step-by-Step Calculation:
- Identify ( n ) and ( r ):
- Here, ( n = 10 ) and ( r = 2 ).
- Calculate ( n! ):
[
10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800
] - Calculate ( r! ):
[
2! = 2 \times 1 = 2
] - Calculate ( (n – r)! ):
[
(10 – 2)! = 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40,320
] - Substitute into the combination formula:
[
10C2 = \frac{10!}{2! \times (10 – 2)!} = \frac{10!}{2! \times 8!}
]
[
= \frac{3,628,800}{2 \times 40,320}
] - Perform the multiplication:
[
2 \times 40,320 = 80,640
] - Now, perform the division:
[
\frac{3,628,800}{80,640} = 45
]
Conclusion
Thus, the value of ( 10C2 ) is ( 45 ).
This means there are 45 different ways to choose 2 items from a set of 10, where the order in which the items are chosen does not matter. This concept is widely used in combinatorics, probability, and statistics to analyze different scenarios, such as lottery combinations, card games, and team selections. Understanding combinations is crucial when determining possible outcomes in various fields, from mathematics to computer science and everyday decision-making.