The Binomial Theorem
      Factorial
      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
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