Combinations: 11 Objects Taken 5 At A Time Formula

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The number of combinations of 11 objects taken 5 at a time can be determined using what formula?

Let's dive into the world of combinations! When we're trying to figure out how many different ways we can select a group of items from a larger set, and the order in which we pick them doesn't matter, we're dealing with combinations. In this case, we want to know how many ways we can choose 5 objects from a set of 11 distinct objects.

Understanding Combinations

The formula for combinations is a cornerstone of combinatorics, a branch of mathematics dealing with counting, arrangement, and combination. It helps us solve problems where we need to select items from a larger set without regard to the order of selection. This is different from permutations, where the order does matter.

The combination formula is expressed as:

nCr = n! / (r! * (n-r)!)

Where:

  • n is the total number of items in the set.
  • r is the number of items we want to choose.
  • ! denotes the factorial, which means multiplying a number by all the positive integers less than it (e.g., 5! = 5 * 4 * 3 * 2 * 1).

Applying the Formula to Our Problem

In our specific problem, we have 11 objects (n = 11) and we want to choose 5 of them (r = 5). Plugging these values into the combination formula, we get:

11C5 = 11! / (5! * (11-5)!)

11C5 = 11! / (5! * 6!)

Now, let's break down the factorials:

  • 11! = 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
  • 5! = 5 * 4 * 3 * 2 * 1 = 120
  • 6! = 6 * 5 * 4 * 3 * 2 * 1 = 720

So, our equation becomes:

11C5 = (11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / (120 * 720)

We can simplify this by canceling out the 6! from the numerator:

11C5 = (11 * 10 * 9 * 8 * 7) / (5 * 4 * 3 * 2 * 1)

11C5 = (11 * 10 * 9 * 8 * 7) / 120

Now, let's do some more simplification. We can divide 10 by 5 to get 2, and 8 by 4 to get 2, and 9 by 3 to get 3, and 2 by 2 to get 1. This simplifies our calculation:

11C5 = (11 * 2 * 3 * 2 * 7) / 1

11C5 = 11 * 2 * 3 * 2 * 7

11C5 = 462

Therefore, there are 462 different ways to choose 5 objects from a set of 11 objects.

Why the Other Options Are Incorrect

Let's quickly look at why the other options provided are not the correct formula:

  • A. 511: This is simply 5 raised to the power of 11, which has nothing to do with combinations.
  • C. 115: This is 11 raised to the power of 5, also unrelated to combinations. It would represent something entirely different, such as the number of ways to assign one of 11 options to each of 5 slots, where repetition is allowed and order matters.
  • B. 5C11: This is incorrect because in combinations, the total number of items (n) must come before the number of items you are choosing (r). You can't choose 11 items from a set of only 5.

Key Differences: Combinations vs. Permutations

It's crucial to distinguish combinations from permutations. As mentioned earlier, combinations are about selecting groups where order doesn't matter. Permutations, on the other hand, are about arrangements where order does matter. The formula for permutations is:

nPr = n! / (n-r)!

Consider an example to illustrate the difference: Suppose we have three letters: A, B, and C. We want to choose two letters.

  • Combinations: If we're looking for combinations, AB is the same as BA because the order doesn't matter. The possible combinations are AB, AC, and BC.
  • Permutations: If we're looking for permutations, AB and BA are different because the order does matter. The possible permutations are AB, BA, AC, CA, BC, and CB.

Real-World Applications of Combinations

Combinations are used extensively in various fields, including:

  • Probability: Calculating the likelihood of winning a lottery or drawing specific cards in a game.
  • Statistics: Designing experiments and analyzing data.
  • Computer Science: Analyzing algorithms and data structures. For example, determining the number of possible subsets of a set.
  • Game Theory: Determining optimal strategies in games.
  • Cryptography: Creating secure codes and ciphers.
  • Genetics: Analyzing genetic combinations and variations.

For instance, imagine you're forming a committee of 3 people from a group of 10. The order in which you select the committee members doesn't matter, so this is a combination problem. You'd use the formula 10C3 to find the number of possible committees.

Tips for Solving Combination Problems

Here are some helpful tips when tackling combination problems:

  1. Identify if order matters: If the order of selection is irrelevant, it's a combination problem. If order matters, it's a permutation problem.
  2. Identify 'n' and 'r': Determine the total number of items (n) and the number of items you're selecting (r).
  3. Apply the formula: Use the combination formula: nCr = n! / (r! * (n-r)!).
  4. Simplify: Simplify the factorial expressions to make calculations easier. Look for opportunities to cancel out terms.
  5. Use a calculator: For larger numbers, a calculator with a factorial function can be very helpful.
  6. Double-check: Review your work to ensure you've correctly identified n and r and that you've applied the formula accurately.

Let's Practice!

Okay, guys, let's put this into practice with a quick example. Suppose you have a pizza place, and they offer 8 different toppings. You want to order a pizza with exactly 3 toppings. How many different topping combinations are possible?

Well, we have 8 total toppings (n = 8), and we want to choose 3 (r = 3). Order doesn't matter (pepperoni, mushrooms, olives is the same as olives, pepperoni, mushrooms!), so this is a combination.

Using our formula:

8C3 = 8! / (3! * (8-3)!) = 8! / (3! * 5!) = (8 * 7 * 6 * 5!)/(3 * 2 * 1 * 5!) = (8 * 7 * 6) / (3 * 2 * 1) = 336 / 6 = 56

So, there are 56 different topping combinations possible.

Conclusion

Therefore, the correct formula to determine the number of combinations of 11 objects taken 5 at a time is 11C5. This formula, rooted in the principles of combinatorics, allows us to efficiently calculate the number of ways to choose a subset of items from a larger set when the order of selection is not important. Understanding and applying this formula is fundamental in various fields, from probability and statistics to computer science and game theory. So, next time you encounter a combination problem, remember the formula and break it down step by step. You've got this!