100 Choose 10

Combination Calculator
Total items (n)
Items chosen (k)

What is 100 choose 10?

Answer

100 choose 10 is
17,310,309,456,440

Order doesn't matter, no repeats: 17,310,309,456,440 ways to choose 10 of 100 items.

C(100, 10) = 100! / (10! · 90!)

= (100 × 99 × 98 × … × 91) / (10 × 9 × 8 × … × 1)

10 factors on top and bottom

= 62,815,650,955,529,472,000 / 3,628,800

= 17,310,309,456,440

If order matters (permutations): P(100, 10) = 100 × 99 × 98 × … × 91 = 62,815,650,955,529,472,000 — each combination in 10! = 3,628,800 orders.

Row 100 of Pascal's triangle
k100 choose k
01
……
716,007,560,800
8186,087,894,300
91,902,231,808,400
1017,310,309,456,440
11141,629,804,643,600
121,050,421,051,106,700
137,110,542,499,799,200
……
1001

See Also

Recalculate with different values ↑

n choose 10 for other n

n choose 10Answer
2 choose 100
10 choose 101
11 choose 1011
12 choose 1066
13 choose 10286
14 choose 101,001
15 choose 103,003
16 choose 108,008
17 choose 1019,448
18 choose 1043,758
19 choose 1092,378
20 choose 10184,756
100 choose 1017,310,309,456,440

FAQ

What is 100 choose 10?

100 choose 10 is 17,310,309,456,440

Why is 100 choose 10 the same as 100 choose 90?

Every way to choose 10 of 100 items leaves exactly 90 behind, so choosing 10 and choosing which 90 to leave out are the same count: 100 choose 10 = 100 choose 90 = 17,310,309,456,440. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(100, 10) = 62,815,650,955,529,472,000. Each of the 17,310,309,456,440 combinations can be put in order in 10! = 3,628,800 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also