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Combination and Permutation Calculator + Online Solver With Free Steps


The Combination and Permutation Calculator finds the possible combinations or grouped permutations given the total items in a set “n” and the number of items taken at a time “k”. You can select between the calculation of combination or permutation through a drop-down menu.

What Is the Combination and Permutation Calculator?

The Combination and Permutation Calculator is an online tool that calculates the number of possible permutations ${}^\mathbf{n}\mathbf{P}_\mathbf{k}$ or combinations ${}^\mathbf{n}\mathbf{C}_\mathbf{k}$ for n items taken k at a time and also displays each combination and permutation as elements in a set.

The calculator interface consists of one drop-down menu labeled “Type” with two options: “Combination” and “Permutation (Grouped).” Here, you select which of the two you want to calculate for your problem.

Additionally, there are two text boxes labeled “Total Items (SET)” and “Items at a time (SUBSET).” The former takes the total number of items (denoted n) or the complete set itself, while the latter specifies how many to take at each step (denoted k).

How To Use the Combination and Permutation Calculator?

You can use the Combination and Permutation Calculator to find the number of possible combinations and permutations for a set by entering the number of items and how many to take at a time. 

For example, suppose you want to find the number of permutations for the following set of natural numbers, taken all at once:

\[ \mathbb{S} = \{ 10,\, 15,\, 20,\, 25,\, 30,\, 35,\, 40 \} \]

The step-by-step guidelines for this are below.

Step 1

Select whether to compute permutation or combination from the drop-down menu “Type.” For the example, you would choose “Permutation (Grouped).”

Step 2

Count the number of items in the set and enter it into the text box “Total Items.” OR, enter the complete set. There are seven total items in the example, so either enter “7” or enter “{10, 15, 20, 25, 30, 35, 40}” without quotes. 

Note: For sets containing words, enclose all words in quotes (see Example 2). 

Step 3

Enter the group of items taken at a time into the text box “Items taken at a time.” To take all of them as in the example, enter “7” without quotes.

Step 4

Press the Submit button to get the results.

Results

The results contain three sections that show under the calculator labeled:

  1. Input Interpretation: The input as the calculator interprets it for manual verification. It categorizes the input as objects and the combination/permutation size.
  2. Number of distinct $\mathbf{k}$ permutations/combinations of $\mathbf{n}$ objects: This is the actual result value for ${}^nP_k$ or ${}^nC_k$ as per the input.
  3. $\mathbf{k}$ permutations/combinations of {set}: All the possible permutations or combinations as distinct elements, with a total count by the end. If the total is exceptionally high, this section is not displayed.

Do note that if you only entered the number of items in the “Total Items” text box (“7” in our example), the third section displays “{1, 2} | {1, 3} | …” instead of the original values. For exactly the values in the input set, enter the full set (see Example 2).

How Does the Combination and Permutation Calculator Work?

The Combination and Permutation Calculator works by using the following equations:

\[ \text{k-permutation} = {}^nP_k = \frac{n!}{(n-k)!} \tag*{$(1)$} \]

\[ \text{k-combination} = {}^nC_k = \frac{n!}{k!(n-k)!} \tag*{$(2)$} \]

Where n and k are non-negative integers (or whole numbers):

\[ n,\, k \in \mathbb{W} = \{0,\, 1,\, 2,\, \ldots\} \wedge k \leq n \]

Factorials

“!” is called the factorial such that $x! = x \times (x-1) \times (x-2) \cdots \times 1$ and 0! = 1. The factorial is defined only for non-negative integers +$\mathbb{Z}$ = $\mathbb{W}$ = {0, 1, 2, …}. 

Since the number of items in a set cannot be a non-integer value, the calculator only expects integers in the input text boxes. 

Difference Between Permutation and Combination

Consider the set:

\[ \mathbb{S} =  \left\{ 1,\, 2,\, 3 \right\} \]

Permutation represents the possible number of arrangements of the set where the order matters. This means that {2, 3} $\neq$ {3, 2}. If the order does not matter (i.e., {2, 3} = {3, 2}), we get the combination instead, which is the number of distinct arrangements.

Comparing equations (1) and (2), the values of C and P are related for a given value of n and k as:

\[  {}^nC_k = \frac{1}{k!} ({}^nP_k) \]

The term (1/k!) removes the effect of the order, resulting in distinct arrangements.

Solved Examples

Example 1

Find the number of combinations of 5 elements at a time possible for the first 20 entries of the set of natural numbers.

Solution

\[ \mathbb{S} = \{ 1,\, 2,\, 3,\, \ldots,\, 20 \} \]

Given that n = 20 and k = 5, equation (1) implies:

\[ {}^{20}C_5(\mathbb{S}) = \frac{20!}{5!(20-5)!} = \frac{20!}{5!(15!)} \]

\[ \Rightarrow \, {}^{20}C_5(\mathbb{S}) = \mathbf{15504} \]

Example 2

For the given set of fruits:

\[ \mathbb{S} =  \left\{ \text{Mangoes},\, \text{Bananas},\, \text{Guavas} \right\} \]

Calculate the combination and permutation for any two fruits taken at a time. Write each combination/permutation distinctly. Further, illustrate the difference between permutation and combination using the results.

Solution

\[ {}^3C_2(\mathbb{S}) = 3 \]

\[ \text{set form} = \big\{ \{ \text{Mangoes},\, \text{Bananas} \},\, \{ \text{Mangoes},\, \text{Guavas} \},\, \{ \text{Bananas},\, \text{Guavas} \} \big\} \]

\[ {}^3P_2(\mathbb{S}) = 6 \]

\[ \text{set form} = \left\{ \begin{array}{rr} \{ \text{Mangoes},\, \text{Bananas} \}, & \{ \text{Bananas},\, \text{Mangoes} \}, \\ \{ \text{Mangoes},\, \text{Guavas} \}, & \{ \text{Guavas},\, \text{Mangoes} \}, \\ \{ \text{Bananas},\, \text{Guavas} \}, & \{ \text{Guavas},\, \text{Bananas} \}\; \end{array} \right\} \]

To get the above results from the calculator, you have to enter “{‘Mangoes, ‘Bananas, ‘Guavas’}” (without double quotes) in the first text box and “2” without quotes in the second.

If you enter “3” in the first box instead, it will still give the correct number of permutations/combinations, but the set form (third section in the results) will be incorrectly displayed.

We can see that the number of permutations is double that of the combinations. Because order does not matter in combinations, each element of the combination set is distinct. That is not the case in permutation, so for a given n and k, we generally have:

\[ {}^nP_k \geq {}^nC_k \] 

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