Binomial sequence

The binomial coefficients C(n, 0), C(n, 1), …, C(n, n) form a sequence that rises to the middle and falls symmetrically. It is row n of Pascal's triangle and adds to 2ⁿ.

C(n, k)Binomial sequence

📐 Formula

C(n, k) = n!/(k!(n − k)!); C(n, k) = C(n, n − k)

✏️ Worked example

n = 4: 1, 4, 6, 4, 1, which adds to 16 = 2⁴.

🌍 Where it is used

Counting outcomes of coin tosses; the binomial probability distribution.

❓ Quick check

C(5, 2) = ?

🏋️ Practice

EasyC(6, 0) = ?

Answer: 1

MediumC(6, 3) = ?

Answer: 20

ChallengeWhy is the sequence symmetric?

Answer: C(n, k) = C(n, n − k)