Find the number of combinations: 5C4
Find the number of combinations: 5C4
The Correct Answer and Explanation is:
Correct Answer:
(54)=5\binom{5}{4} = 5
Explanation
The expression 5C4, also written as (54)\binom{5}{4}, represents the number of combinations of 5 items taken 4 at a time. In combinatorics, a combination refers to the selection of items without regard to the order in which they are selected. This is different from a permutation, where order does matter.
Formula for Combinations:
(nr)=n!r!(n−r)!\binom{n}{r} = \frac{n!}{r!(n – r)!}
Where:
- nn = total number of items
- rr = number of items to choose
- n!n! (n factorial) = n×(n−1)×(n−2)×…×1n \times (n – 1) \times (n – 2) \times \ldots \times 1
Applying to 5C4:
(54)=5!4!(5−4)!=5!4!⋅1!\binom{5}{4} = \frac{5!}{4!(5 – 4)!} = \frac{5!}{4! \cdot 1!}
Calculating factorials:
- 5!=1205! = 120
- 4!=244! = 24
- 1!=11! = 1
(54)=12024⋅1=12024=5\binom{5}{4} = \frac{120}{24 \cdot 1} = \frac{120}{24} = 5
Interpretation:
Choosing 4 items out of 5 without caring about order gives us 5 unique combinations. This makes intuitive sense. If you have 5 people — say A, B, C, D, and E — and you want to select 4 of them, then the number of ways you can do that is equal to the number of different people who could be left out (since selecting 4 out of 5 is equivalent to omitting 1 out of 5). You can leave out A, or B, or C, or D, or E — 5 possibilities.
Conclusion:
(54)=5\boxed{\binom{5}{4} = 5}
This is a basic but important result in combinatorics, useful in probability, statistics, and many areas of mathematics and computer science.
