The binomial theorem allows us to expand expressions of the form ((x + y)^n). The coefficients in the expansion are derived using the binomial coefficient, which is written as (
). Let’s break this down step by step.
What does (
mean?
The binomial coefficient ( tells us:
- How many ways we can arrange (k) (y)’s and (n-k) (x)’s in the expansion of ((x + y)^n).
The formula is:
(
Where:
- (
is the factorial of (n) (the product of all integers from 1 to (n)),
- (
is the factorial of (k), and
is the factorial of
.
Let’s see how this works with examples.
Example 1: Expanding ((x + y)^3)
When expanding ((x + y)^3), we want to find all the possible ways to arrange 3 terms, where:
- (k) terms are (y),
- (n-k = 3-k) terms are (x).
Case 1: (k = 0) (All (x)’s, no (y)’s)
The arrangement is (xxx).
There is only 1 way to arrange this:
(
Case 2: (k = 1) (1 (y) and 2 (x)’s)
The arrangements are: (xyx, xxy, yxx).
There are 3 ways to arrange these:
(
Case 3: (k = 2) (2 (y)’s and 1 (x))
The arrangements are: (xyy, yxy, yyx).
There are 3 ways to arrange these:
(
Case 4: (k = 3) (All (y)’s, no (x)’s)
The arrangement is (yyy).
There is only 1 way to arrange this:
(
Example 2: Expanding ((x + y)^4)
Now, let’s expand ((x + y)^4). This means arranging 4 terms, where:
- (k) terms are (y),
- (n-k = 4-k) terms are (x).
Case 1: (k = 0) (All (x)’s, no (y)’s)
The arrangement is (xxxx).
There is only 1 way:
(
Case 2: (k = 1) (1 (y) and 3 (x)’s)
The arrangements are: (xyxx, xxyx, xxxy, yxxx).
There are 4 ways:
(
Case 3: (k = 2) (2 (y)’s and 2 (x)’s)
The arrangements are: (xxyy, xyxy, xyyx, yxxy, yxyx, yyxx).
There are 6 ways:
(
Case 4: (k = 3) (3 (y)’s and 1 (x))
The arrangements are: (yyyx, yyyx, yxyy, xyyy).
There are 4 ways:
(
Case 5: (k = 4) (All (y)’s, no (x)’s)
The arrangement is (yyyy).
There is only 1 way:
(
Key Idea
Each ( value tells us the number of unique arrangements of (k) (y)’s and (n-k) (x)’s in the binomial expansion. This is why:
(
By summing up all possible arrangements for every (k), we get the full expansion.
Visualization with Pascal’s Triangle
The coefficients in Pascal’s Triangle represent these arrangements:
Each number in the row corresponds to a ( value.
Conclusion
The binomial theorem provides a powerful way to expand expressions like ((x + y)^n). The coefficients ( count the number of ways to arrange (y)’s and (x)’s, and understanding this process can unlock deeper insights into combinatorics and algebra.
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