The total number of ways to choose 4 marbles from 16 is: - Noxie

Understanding the Context

Understanding the Problem

At first glance, choosing 4 marbles from 16 might seem like a straightforward arithmetic problem. However, the key distinction lies in whether the order of selection matters:

• If order matters, you’re dealing with permutations — calculating how many ways marbles can be arranged when position is important.
  - But if order doesn’t matter, and you only care about which marbles are selected (not the sequence), you’re looking at combinations.

Since selecting marbles for a collection typically concerns selection without regard to order, we focus on combinations — specifically, the number of combinations of 16 marbles taken 4 at a time, denoted mathematically as:

Key Insights

$$
\binom{16}{4}
$$

What is a Combination?

A combination is a way of selecting items from a larger set where the order of selection is irrelevant. The formula to compute combinations is:

$$
\binom{n}{r} = \frac{n!}{r!(n - r)!}
$$

Final Thoughts

Where:
- $ n $ = total number of items (here, 16 marbles)
- $ r $ = number of items to choose (here, 4 marbles)
- $ ! $ denotes factorial — the product of all positive integers up to that number (e.g., $ 5! = 5 \ imes 4 \ imes 3 \ imes 2 \ imes 1 = 120 $)

Using this formula:

$$
\binom{16}{4} = \frac{16!}{4!(16 - 4)!} = \frac{16!}{4! \cdot 12!}
$$

Note that $ 16! = 16 \ imes 15 \ imes 14 \ imes 13 \ imes 12! $, so the $ 12! $ cancels out:

$$
\binom{16}{4} = \frac{16 \ imes 15 \ imes 14 \ imes 13}{4 \ imes 3 \ imes 2 \ imes 1}
$$