A factorial function is a function that multiplies first natural numbers. For a natural number with we denote the multiplication of first natural numbers.
In addition, it is useful to define the value We define by agreement.
By inductive conclusion, we can see that the factorials match the formula
with initial value
Example 1. Calculate the following expression:
As a tip, when dealing with factorials never jump ahead and calculate everything because you can always somehow use the recursion. Now, since we know that we have a common factor of all terms:
Example 2. Solve the following equation:
is the only solution to the default equation because must be a natural number.
Let both and are natural numbers and if can also have the value of and The binomial coefficient is denoted with symbol and defined as:
for For by definition, we have:
When replacing with in the definition of the binomial coefficient we get:
The symmetry property
Example 3. Calculate using the symmetry property as above.
Let’s count the binomial coefficients for small value numbers and their calculated values as shown below:
The triangle above is commonly known as a Pascal’s or Chinese triangle. Write the calculated values with two more added lines:
Each element from the Pascal’s triangle is equal to the addition of two elements on the line above on the right and left side of applicable element, except the constant value of the elements on the edges of the triangle which are always equal to
This property is valid for any element of Pascal’s triangle. Choose three characteristic elements of Pascal’s triangle:
The following relation describes the basic principle of Pascal’s triangle:
It has been proven that the left side of the equal sign is equal to the right.
This property demonstrates that all binomial coefficients are natural numbers because each of them is equal to or to the sum of two natural numbers.
The binomial theorem
The binomial theorem, is also known as binomial expansion, which explains the expansion of powers. It only applies to binomials. Let’s take a look at the link between values in Pascal’s triangle and the display of the powers of the binomial
For small values of a natural number we know the formulas for the powers of the binomial:
The last two formulas are called square of a binomial and cube of a binomial, respectively. The coefficients in these formulas are in fact binomial coefficients.
If we continue further, we get:
Notice that all addends come in the form . Numbers are called binomial coefficients. They are easily calculated and noted using factorials.
The binomial theorem
For is valid:
Example 4. Through using the above formula expand the following: