Combination Calculator nCr – Choose Formula

Combinations count selections where order does not matter. Choosing 2 students from 5 gives C(5,2) = 5!/(2!×3!) = 120/12 = 10 pairs. The factorial expansion cancels nicely for hand work: C(10,3) = (10×9×8)/(3×2×1) = 720/6 = 120. Contrast with permutations, where order matters: P(5,2) = 20 because each pair {A,B} splits into AB and BA — the ratio is exactly r!, here 2. The calculator prints the canceled fraction, the decimal value and the nPr comparison together. Lottery odds make the scale vivid: C(49,6) = 13,983,816 distinct tickets, which is why a single line has about a 1-in-14-million jackpot chance. Values update live as you adjust n and r, with symmetry C(n,r) = C(n,n−r) applied to keep arithmetic small.

1. 1

   Enter n and r

   Set the pool size n and how many you choose, r ≤ n

2. 2

   Expand the factorials

   The tool shows n! over r!(n−r)! with the canceled shortcut

3. 3

   Compute C(n,r)

   Get the exact count of unordered selections

4. 4

   Compare with nPr

   See P(n,r) = C(n,r) × r! side by side

• 1

  C(n,r) = C(n,n−r): choosing 2 winners equals choosing 3 losers from 5

• 2

  Order matters? Use nPr. Order irrelevant? Use nCr

• 3

  The canceled form (n×…×(n−r+1))/r! avoids giant factorials by hand

• Using permutations when order is irrelevant, inflating C(5,2)=10 to 20

• Forgetting r! in the denominator and getting 60 for C(5,2)

• Mixing up n and r — C(2,5) is impossible since r cannot exceed n

• Double counting subsets when generating them by hand