Binomial Coefficients for Numeric and Symbolic Arguments

Compute the binomial coefficients for these expressions.

syms n
[nchoosek(n, n), nchoosek(n, n + 1), nchoosek(n, n - 1)]

ans =
[ 1, 0, n]

If one or both parameters are negative numbers, convert these numbers to symbolic objects.

[nchoosek(sym(-1), 3), nchoosek(sym(-7), 2), nchoosek(sym(-5), -5)]

ans =
[ -1, 28, 1]

If one or both parameters are complex numbers, convert these numbers to symbolic objects.

[nchoosek(sym(i), 3), nchoosek(sym(i), i), nchoosek(sym(i), i + 1)]

ans =
[ 1/2 + 1i/6, 1, 0]

Handle Expressions Containing Binomial Coefficients

Many functions, such as diff and expand, can handle expressions containing nchoosek.

Differentiate the binomial coefficient.

syms n k
diff(nchoosek(n, 2))

ans =
-(psi(n - 1) - psi(n + 1))*nchoosek(n, 2)

Expand the binomial coefficient.

expand(nchoosek(n, k))

ans =
-(n*gamma(n))/(k^2*gamma(k)*gamma(n - k) - k*n*gamma(k)*gamma(n - k))

Pascal Triangle

Use nchoosek to build the Pascal triangle.

m = 5;
for n = 0:m
    C = sym([]);
  for k = 0:n
    C = horzcat(C, nchoosek(n, k));
  end
  disp(C)
end

1
[ 1, 1]
[ 1, 2, 1]
[ 1, 3, 3, 1]
[ 1, 4, 6, 4, 1]
[ 1, 5, 10, 10, 5, 1]

All Combinations of Vector Elements

Find all combinations of elements of a 1-by-5 symbolic row vector taken three and four at a time.

Create a 1-by-5 symbolic vector with the elements x1, x2, x3, x4, and x5.

v = sym('x', [1, 5])

v =
[ x1, x2, x3, x4, x5]

Find all combinations of the elements of v taken three at a time.

C = nchoosek(v, 3)

C =
[ x1, x2, x3]
[ x1, x2, x4]
[ x1, x3, x4]
[ x2, x3, x4]
[ x1, x2, x5]
[ x1, x3, x5]
[ x2, x3, x5]
[ x1, x4, x5]
[ x2, x4, x5]
[ x3, x4, x5]

C = nchoosek(v, 4)

C =
[ x1, x2, x3, x4]
[ x1, x2, x3, x5]
[ x1, x2, x4, x5]
[ x1, x3, x4, x5]
[ x2, x3, x4, x5]