How to Calculate Combinations?

Combinations are a way to select items from a larger pool where the order does not matter. For instance, if you have a deck of cards and you want to choose 5 cards, the order in which you pick them doesn’t matter. This is different from permutations, where order does matter.

The Formula for Combinations

The formula to calculate combinations is:

$C(n, k) = frac{n!}{k!(n-k)!}$

Here, $n$ is the total number of items, and $k$ is the number of items to choose. The exclamation mark (!) denotes a factorial, which means multiplying a series of descending natural numbers. For example, $5! = 5 times 4 times 3 times 2 times 1 = 120$

Step-by-Step Calculation

Let’s break down the steps of calculating combinations with an example. Suppose you have 10 books and you want to choose 3 out of them.

  1. Identify $n$ and $k$: In this case, $n = 10$ and $k = 3$
  2. Calculate $n!$: $10! = 10 times 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1 = 3,628,800$
  3. Calculate $k!$: $3! = 3 times 2 times 1 = 6$
  4. Calculate $(n – k)!$: $(10 – 3)! = 7! = 7 times 6 times 5 times 4 times 3 times 2 times 1 = 5,040$
  5. Plug these values into the formula:

$C(10, 3) = frac{10!}{3!(10-3)!} = frac{3,628,800}{6 times 5,040} = frac{3,628,800}{30,240} = 120$

So, there are 120 ways to choose 3 books out of 10.

Practical Example

Imagine you are organizing a committee of 4 members from a group of 12 people. To find out how many different committees can be formed, you would use the combination formula.

  1. Identify $n$ and $k$: Here, $n = 12$ and $k = 4$
  2. Calculate $n!$: $12! = 479,001,600$
  3. Calculate $k!$: $4! = 24$
  4. Calculate $(n – k)!$: $8! = 40,320$
  5. Plug these values into the formula:

$C(12, 4) = frac{12!}{4!(12-4)!} = frac{479,001,600}{24 times 40,320} = frac{479,001,600}{967,680} = 495$

Therefore, there are 495 different ways to form a committee of 4 members from a group of 12 people.

Conclusion

Understanding how to calculate combinations can be extremely useful in various fields, from mathematics and statistics to everyday decision-making. The key is to remember that order does not matter in combinations, and using the formula $C(n, k) = frac{n!}{k!(n-k)!}$ makes the calculation straightforward. Happy calculating!

Citations

  1. 1. Khan Academy – Combinations
  2. 2. Math is Fun – Combinations and Permutations
  3. 3. Purplemath – Combinations

Related

(2) O3 + H → O2 + OH k2 = 1.78×10^-11 cm^3 s^-1 (3) O + OH → O2 + H k3 = 4.40×10^-11 cm^3 s^-1 (5) O + HO2 → O2 + OH k5 = 3.50×10^-11 cm^3 s^-1 (6) H + HO2 → O2 + H2 k6 = 5.40×10^-12 cm^3 s^-1 (9) OH + HO2 → O2 + H2O2 k9 = 4.00×10^-11 cm^3 s^-1 (10) HO2 + HO2 → O2 + H2O2 k10 = 2.50×10^-12 cm s^-1 (11) O + O2 + M → O3 + M k11 = 1.05×10^-34 cm^6 s^-1 (14) H + O2 + M → HO2 + M k14 = 8.08×10^-32 cm^6 s^-1 (15) H + H + M → H2O + M k15 = 3.31×10^-27 cm^6 s^-1 (16) O2 + hv → 2 O k16 = (1.26×10^-8 s^-1) φ (17) H2O + hv → H + OH k17 = (3.4×10^-6 s^-1) φ (18) O3 + hv → O2 + O k18 = (7.10×10^-5 s^-1) φ

Table 1 Reactions, rate constants and activation energies used in the model* No. Reaction kopt (M⁻¹ s⁻¹) 1 OH + H₂ → H + H₂O 3.74 x 10⁷ 2 OH + HO₂ → HO₂ + OH⁻ 5 x 10⁹ 3 OH + H₂O₂ → HO₂ + H₂O 3.8 x 10⁷ 4 OH + O₂ → O₂ + OH 9.96 x 10⁹ 5 OH + HO₂ → O₂ + H₂O 7.1 x 10⁹ 6 OH + OH → H₂O₂ 5.3 x 10⁹ 7 OH + e⁻aq → OH⁻ 3 x 10¹⁰ 8 H + O₂ → HO₂ 2.0 x 10¹⁰ 9 H + HO₂ → H₂O₂ 2.0 x 10¹⁰ 10 H + H₂O₂ → OH + H₂O 3.44 x 10⁷ 11 H + OH → H₂O 1.4 x 10¹⁰ 12 H + H → H₂ 1.94 x 10¹⁰ 13 e⁻aq + O₂ → O₂⁻ 1.9 x 10¹⁰ 14 e⁻aq + O₂ → HO₂⁻ + OH⁻ 1.3 x 10¹⁰ 15 e⁻aq + HO₂ 2.0 x 10¹⁰ 16 e⁻aq + H₂O₂ 1.1 x 10¹⁰ 17 e⁻aq + HO₂ → OH + OH⁻ 1.3 x 10¹⁰ 18 e⁻aq + H⁺ → H 2.3 x 10¹⁰ 19 e⁻aq + e⁻aq → H₂ + OH⁻ + OH⁻ 2.5 x 10⁹ 20 HO₂ + O₂ → O₂ + HO₂ 1.3 x 10⁹ 21 HO₂ + HO₂ → O₂ + H₂O₂ 8.3 x 10⁵ 22 HO₂ + HO₂ → O₂ + OH + H₂O 3.7 23 HO₂ + HO₂ → O₂ + O₂ + OH + H₂O 7 x 10⁵ s⁻¹ 24 H⁺ + O₂⁻ → HO₂ 4.5 x 10¹⁰ 25 H⁺ + O₂⁻ → O₂ 2.0 x 10¹⁰ 26 H⁺ + OH⁻ 1.4 x 10¹¹ 27 H⁺ + HO₂⁻ 2 x 10¹⁰ 28 H₂O₂ → HO₂ + H⁺ + OH⁻ 2.5 x 10⁻⁵ s⁻¹ 29 H₂O₂ → H⁺ + OH⁻ 1.4 x 10⁻⁷ s⁻¹ 30 O₂ + O₂ → O₂ + HO₂ + OH⁻ 0.3 31 O₂ + H₂O₂ → O₂ + OH + OH 16 32

(2) O3 + H → O2 + OH k2 = 1.78×10^-11 cm^3 s^-1 (3) O + OH → O2 + H k3 = 4.40×10^-11 cm^3 s^-1 (5) O + HO2 → O2 + OH k5 = 3.50×10^-11 cm^3 s^-1 (6) H2O + O → 2 OH k6 = 5.40×10^-12 cm^3 s^-1 (9) OH + HO2 → O2 + H2O k9 = 4.00×10^-11 cm^3 s^-1 (10) HO2 + HO2 → O2 + H2O2 k10 = 2.50×10^-12 cm s^-1 (11) O + O2 + M → O3 + M k11 = 1.05×10^-34 cm^6 s^-1 (14) H + O2 + M → HO2 + M k14 = 8.08×10^-32 cm^6 s^-1 (15) OH + H + M → H2O + M k15 = 3.31×10^-27 cm^6 s^-1 (16) O2 + hv → 2 O k16 = (1.26×10^-8 s^-1) φ (17) H2O + hv → H + OH k17 = (3.4×10^-6 s^-1) φ (18) O3 + hv → O2 + O k18 = (7.10×10^-8 s^-1) φ