91Ó°ÊÓ

When it comes to selecting a group of items where the order doesn't matter, we use combinations. In the scenario where an adviser chooses 4 students from a class of 12 to perform the same task, we're dealing with combinations.The formula for combinations is given by:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]where:

• \(n\) is the total number of items to choose from (in our example, 12 students)
• \(k\) is the number of items to choose (in our example, 4 students)

Using this formula, the number of ways to choose 4 students out of 12 is:\[ \binom{12}{4} = \frac{12!}{4! \times 8!} = 495 \]Here, the factorial operation \(!\) denotes the product of all positive integers up to the given number, making the calculation feasible and systematic for any similar problem.

Permutations are all about order. When students are assigned different tasks, the order in which they are assigned matters, and thus permutations come into play.In a scenario where each selected student from the group receives a unique task, the formula stretches to accommodate order:After choosing 4 students from 12, these students can be assigned tasks in permutations of themselves.The general formula for permutations when selecting \(k\) items from \(n\) is:\[ P(n, k) = \frac{n!}{(n-k)!} \]However, in our exercise, once the 4 students (from the combination \(\binom{12}{4} = 495\)) are chosen, each student can be assigned one of the 4 unique tasks. This is calculated simply as:\[ 4! = 24 \]So, the number of ways is:\[ \binom{12}{4} \times 4! = 495 \times 24 = 11880 \]The combination of selecting the group and then permuting their order makes permutation greatly powerful whenever specifics of order count.

The term 'factorial' is key to comprehending permutations and combinations comprehensively. Factorial of a non-negative integer \(n\), denoted as \(n!\), represents the product of all positive integers up to \(n\).For instance:

• \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)
• \(3! = 3 \times 2 \times 1 = 6\)
• Factorial of zero is 1, \(0! = 1\)

Factorial forms the backbone of computations involving permutations and combinations.In our exercise, it calculates the number of possible arrangements of a selected group or the number of ways to choose a smaller group from a large set. The factorial grows rapidly with larger numbers, making it essential for determining possibilities whenever grouping or ordering is in play.