text: # Combinations Calculator (nCr)
  
  Enter the number of elements (n) and the sample number (r).
  
  n choose r, $$C(n,r)={n\choose r}$$.
  
  #### Step-by-step solution:
  
  Use this calculator to find the result of a combination. Enter the total number of items and the number of items chosen.
  
  ## How to use the combination calculator?
  
  Step 1: Enter the total number of elements in the first input box. This is the value of n.
  
  Step 2: Enter the number of elements chosen from the set in the second input box. This is the value of r.
  
  Step 3: Click “Calculate” to get the result of the combination.
  
  Step 4: The answer will be displayed on the right and the step-by-step solution will be displayed at the bottom.
  
  ## What types of numbers can I enter into the calculator?
  
  The values of n and r must be positive and integers. Since we are talking about sets of elements, we cannot have negative or fractional numbers to calculate combinations.
  
  Also, the value of r must be less than or equal to the value of n. The value of n is the total number of elements and the value of r is the number of elements chosen, so r cannot be greater than n.
  
  Therefore, taking into account these two conditions, the entered numbers must follow the following:
  
  $$n\geq r\geq 0$$
  
  ## What are combinations?
  
  A combination is a mathematical technique that determines the number of possible arrangements of elements, where the order does not matter. In combinations, we can select the elements in any order.
  
  ## How to find combinations?
  
  Combinations can be found using the combinations’ formula. This formula determines the number of possible ways to select only a few objects from a set without repetition:
  
  $$C(n,~r)={n \choose r}=\frac{n!}{r!(n-r)!}$$
  
  where n is the total number of elements in the set and r is the number of chosen objects.
  
  For example, if we have a set of 8 elements and we choose 6, we can calculate the number of combinations as follows:
  
  $$C( n,~r)=\frac{n!}{r!(n-r)!}$$
  
  $$C( 8,~6)=\frac{8!}{6!(8-6)!}$$
  
  $$=\frac{8!}{6!(2)!}$$
  
  Now, we can rewrite to 8! like 8×7×6!, to then simplify:
  
  $$C( 8,~6)=\frac{8\times 7 \times 6!}{6!(2)!}$$
  
  $$=\frac{8\times 7 }{2!}$$
  
  $$=28$$
  
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