How many 3/4 are contained in 24?
How many 3/4 are contained in 24
The Correct Answer and Explanation is:
Correct Answer:
$$
\binom{5}{4} = 5
$$
Explanation
The expression 5C4, also written as $\binom{5}{4}$, 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:
$$
\binom{n}{r} = \frac{n!}{r!(n – r)!}
$$
Where:
- $n$ = total number of items
- $r$ = number of items to choose
- $n!$ (n factorial) = $n \times (n – 1) \times (n – 2) \times \ldots \times 1$
Applying to 5C4:
$$
\binom{5}{4} = \frac{5!}{4!(5 – 4)!} = \frac{5!}{4! \cdot 1!}
$$
Calculating factorials:
- $5! = 120$
- $4! = 24$
- $1! = 1$
$$
\binom{5}{4} = \frac{120}{24 \cdot 1} = \frac{120}{24} = 5
$$
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:
$$
\boxed{\binom{5}{4} = 5}
$$
This is a basic but important result in combinatorics, useful in probability, statistics, and many areas of mathematics and computer science.
