Binomial coefficients
The binomial theorem, sigma notation and binomial expansion algorithm
The binomial theorem and binomial expansion algorithm examples
The Binomial Theorem
Factorial
The factorial is defined for a positive integer n, denoted n! represents the product of all positive integers less than or equal to n
n! = n · (n - 1) ·  ·  · 2 · 1.
The first few factorials are, 1! = 1,   2! = 2 · 1 = 23! = 3 · 2 · 1 = 64! = 4 · 3 · 2 · 1 = 24, and so on.
By the definition,     0! = 1.
So for example, n! shows the number of ordered arrangements or permutations of n objects, that is, on how many ways n distinct objects can be arranged in a row.
Thus, for example four digits  1, 2, 3, 4 can be arranged in 4! = 24  ways, as is shown below
1, 2, 3, 4         2, 1, 3, 4         3, 1, 2, 4         4, 1, 2, 3
1, 2, 4, 3         2, 1, 4, 3         3, 1, 4, 2         4, 1, 3, 2
1, 3, 2, 4         2, 3, 1, 4         3, 2, 1, 4         4, 2, 1, 3
1, 3, 4, 2         2, 3, 4, 1         3, 2, 4, 1         4, 2, 3, 1
1, 4, 2, 3         2, 4, 1, 3         3, 4, 1, 2         4, 3, 1, 2
1, 4, 3, 2         2, 4, 3, 1         3, 4, 2, 1         4, 3, 2, 1
Binomial coefficients
A binomial coefficient is a numerical factor that multiply the successive terms in the expansion of the binomial (a + b)n, for integral n, written
So that, the general term, or the (k + 1)th term, in the expansion of (a + b)n,
For example,
A binomial coefficient equals the number of ways that r objects can be selected from n objects without regard to order, called combinations and noted C(n, r) or Cnr.
For example, the number of distinct combinations of three digits selected from  1, 2, 3, 4, 5 is
 1 2 3      2 3 4      3 4 5 1 2 4      2 3 5 1 2 5      2 4 5 1 3 4 1 3 5 1 4 5
The binomial theorem, sigma notation and binomial expansion algorithm
The theorem that shows the form of the expansion of any positive integral power of a binomial (a + b)n to a polynomial with n + 1 terms,
 Example:  Find the middle term of the binomial expansion
 Solution:
 Example:  Find the 7th term of the binomial expansion if the coefficient of the third term
relates to the coefficient of the second term as 9 : 2.
 Solution:
 Example:  Which term of the binomial expansion is missing x?
 Solution:
To fulfill the required condition, the exponent of x must be zero, therefore
Intermediate algebra contents