What is the formula for binomial theorem?

According to the binomial theorem, (a + b)^n is the sum of terms where for each value of k from 0 to n C(n, k) is multiplied by a^(n-k) and then multiplied by b^k. Each coefficient C(n, k) is a binomial coefficient, and the sum of the exponents of a and b in each term is n.

How many terms are there in a binomial expansion?

The expansion of (a+b)^n contains exactly n+1 terms. So (a+b)^2 has 3 terms, (a+b)^5 has 6 terms and so on.

How do you find a particular term in development?

We use the general formula. The term at position r+1 is given by T=C(n,r)*a^(n - r)*b^r. The index r of the first term starts at 0. If you open the option for a particular term and enter the desired position, the calculator will return only that one term.

How does Pascal's triangle give us the coefficients?

The coefficients of the expansion of (a+b)^n are given in row n of Pascal's triangle, starting with C(n,0) through to C(n,n). Each number is the sum of the two numbers above it. For example, the fourth row is 1,4,6,4,1 and these are the coefficients of the expansion of (a+b)^4.

What is the sum of coefficients in a binomial expansion?

When a = b = 1, all the powers are equal to 1, so each term reduces to the coefficient itself and their sum is the nth power of 2. For example, adding the coefficients 1, 4, 6, 4, and 1 from (a + b) to the power 4 gives 16.

How is an expression like (a - b)^n expanded?

You can think of this as (a + (-b))^n. The expansion is the same but the signs alternate and the odd powers of b are negative. A calculator will process the signs in the table for each term when you input a negative value for b.

Binomial Expansion Calculator

Enter the binomial

First term (a)

Second term (b)

Power (n)

Value of (a + b)ⁿ

Find a specific term

Use the general term formula to jump straight to any single term of the expansion.

Value of (a + b)ⁿ

(a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴

With a = 2 and b = 1: 16 + 32 + 24 + 8 + 1 = 81

Number of terms
Sum of the coefficients (2ⁿ)
Largest coefficient

The coefficients read the same forwards and backwards, because C(n, k) = C(n, n − k). The largest one here is 6.

Add every coefficient and you get 16, which is 2 to the power 4.

Full expansion, term by term

Every term of (a + b) to the power n: its coefficient C(n, k), the powers of a and b, and the value once a and b are substituted (shown for n up to 20).
Term	Coefficient C(n, k)	Power of a	Power of b	Term value
1	1	4	0	16
2	4	3	1	32
3	6	2	2	24
4	4	1	3	8
5	1	0	4	1

Pascal's triangle and coefficient chart

Show Pascal's triangle

Display the triangle whose rows are the binomial coefficients used as the coefficients.

Show the coefficient chart

Plot the coefficient C(n, k) for every term to see its symmetric bell shape.

Pascal's triangle. Row n holds the coefficients of (a + b) to the power n; every number is the sum of the two above it (rows up to 12 shown).
Row	k = 0	k = 1	k = 2	k = 3	k = 4
n = 0	1
n = 1	1	1
n = 2	1	2	1
n = 3	1	3	3	1
n = 4	1	4	6	4	1

The binomial expansion is the representation of a sum of two terms (e.g. (a + b) to the power n) as a series of independent terms. Instead of multiplying brackets together repeatedly, the binomial theorem allows you to calculate all the terms at once with the coefficients for each term being directly derived from Pascal's triangle. By entering a, b and the exponent n in the fields above, this calculator will give you the full expansion, a tabular representation of the individual terms, the result of the calculation as well as the general term at any given position.

The binomial theorem states that:

For any nonnegative integer n, the theorem is expressed as follows.

(a + b)^n = \sum_{k=0}^{n} \binom{n}{k}\, a^{\,n-k}\, b^{\,k}

The symbol C(n, k) is read as "n choose k" and denotes the binomial coefficient. It indicates how many brackets of n brackets are needed to contribute to term b. Therefore, the same number appears both in combinatorial problems and in algebra.

If you look at the formula term by term, the exponent of a starts out as n and decreases by one each subsequent term. The exponent of b starts out as 0 and increases gradually up to n. Also, the sum of the two exponents in any given term is equal to n.

How many terms are there and what are the coefficients?

The expansion of (a+b)^n always contains n+1 terms. The coefficients are the numbers in the nth row of Pascal's triangle, where each number is the sum of the two numbers directly above it.

For example, the elements of the fourth row of a triangle are 1, 4, 6, 4, 1, so it looks like this:

(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4

The coefficients are symmetric, meaning that C(n, k) = C(n, n − k), which implies that the nth row of Pascal's triangle is symmetrical. The sum of all the entries of row n equals 2^n. This follows from setting both a and b to 1 in the binomial theorem.

An example with numbers:

We are looking at the power (2x + 3) to the third. In this case a = 2x, b = 3, n = 3, and the third row of Pascal's triangle is 1, 3, 3, 1.

(2x + 3)^3 = 8x^3 + 36x^2 + 54x + 27

Each coefficient is the product of the corresponding Pascal number with the respective powers of 2 and 3. The second term is 3 × (2x)² × 3 = 36x², and the constant term is 1 × 3³ = 27. Regardless of what values are entered for a and b, the calculator will perform the same substitution and show the value of each term.

Power n	Expansion of (a + b) to the power n
0	1
1	a + b
2	a² + 2ab + b²
3	a³ + 3a²b + 3ab² + b³
4	a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴
5	a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + b⁵

Subtraction: Expansion of (a-b)^n

The difference can be considered as a sum with the second term negative. If b is negative, then the signs alternate in the expansion. This is because all odd powers of negative numbers are negative.

(x - 1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1

A calculation tool can easily enter a negative value for b to extend the difference. A table of individual terms instead deals with the signs of each term.

Method for determining individual terms without full expansion:

The term at position r+1 (where the first term is counted as being at position 1) is given by:

T_{r+1} = \binom{n}{r}\, a^{\,n-r}\, b^{\,r}

The fifth term of the sixth power of ((2)x - (5)y) is r = 4 and it looks like this: C(6, 4) × (2x)² × (-5y)⁴ = 15 × 4x² × 625y⁴ = 37,500 x²y⁴. If you select the option for this specific term and enter the required values, the calculator will provide this term.

About the middle terms:

Since the coefficients first increase to a maximum and then decrease, the middle term in the expansion has the largest coefficient. If n is even, there is only one middle term, which occurs at position n/2 + 1. If n is odd, there are two middle terms with equal size. The maximum coefficient shown by the calculator is exactly this middle value, C(n, n/2), rounded to an integer.

On the applications of binomial development:

The expansion forms a bridge between algebra and probability. The probability of getting exactly k successes in n independent trials is the term in the expansion of (p + q)^n, where p and q are the probabilities of success and failure respectively. This forms the basis for the binomial distribution.

Physicists and engineers also use the first terms to get quick approximations. When the change is small, (1+x) raised to the n power is approximately equal to 1+nx, which turns complex powers or roots into simple linear estimates. The same coefficients also appear in calculus, combinatorics, and anywhere else that Pascal's triangle appears.

Frequently asked questions

What is the formula for binomial theorem?: According to the binomial theorem, (a + b)^n is the sum of terms where for each value of k from 0 to n C(n, k) is multiplied by a^(n-k) and then multiplied by b^k. Each coefficient C(n, k) is a binomial coefficient, and the sum of the exponents of a and b in each term is n.
How many terms are there in a binomial expansion?: The expansion of (a+b)^n contains exactly n+1 terms. So (a+b)^2 has 3 terms, (a+b)^5 has 6 terms and so on.
How do you find a particular term in development?: We use the general formula. The term at position r+1 is given by T=C(n,r)*a^(n - r)*b^r. The index r of the first term starts at 0. If you open the option for a particular term and enter the desired position, the calculator will return only that one term.
How does Pascal's triangle give us the coefficients?: The coefficients of the expansion of (a+b)^n are given in row n of Pascal's triangle, starting with C(n,0) through to C(n,n). Each number is the sum of the two numbers above it. For example, the fourth row is 1,4,6,4,1 and these are the coefficients of the expansion of (a+b)^4.
What is the sum of coefficients in a binomial expansion?: When a = b = 1, all the powers are equal to 1, so each term reduces to the coefficient itself and their sum is the nth power of 2. For example, adding the coefficients 1, 4, 6, 4, and 1 from (a + b) to the power 4 gives 16.
How is an expression like (a - b)^n expanded?: You can think of this as (a + (-b))^n. The expansion is the same but the signs alternate and the odd powers of b are negative. A calculator will process the signs in the table for each term when you input a negative value for b.

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References

Wikipedia: Binomial theorem
Statement of the theorem, the general term, and worked expansions.
Wolfram MathWorld: Binomial Theorem
Formal reference on the binomial theorem and its coefficients.
Encyclopaedia Britannica: Binomial theorem
Background on the theorem and its connection to Pascal's triangle.