The Binomial Theorem
      Factorial
      Binomial coefficients and Pascal's triangle
         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 and Pascal's triangle
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, 
The binomial coefficients can also be obtained by using Pascal's triangle.
The triangular array of integers, with 1 at the apex, in which each number is the sum of the two numbers above it in the preceding row, as is shown in the initial segment in the diagram, is called Pascal's triangle. 
The binomial coefficients refer to the nth row, kth element in Pascal's triangle as shows the right diagram. 
So, for example the 6th row of the triangle contains the sequence of the coefficients of a binomial of the 6th power.
n 1
1 1 1
2 1 2 1
3 1 3 3 1
4 1 4 6 4 1
5 1 5 10 10 5 1
6 1 6 15 20 15 6 1
- 0th - 1st - 2nd - 3rd - 4th - 5th - 6th 1
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 and binomial expansion algorithm examples
Example:
Example:  Find the 4th term of the binomial expansion (a - x)5.
Solution:
Or, we can use the formula to find (k + 1)th term.
Since, n = 5  and   k + 1 = 4  =>  k = 3  then by plugging these values into
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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