Binomial expansion specific term calculator
The third term of (2x − 3y)^5 is 720x^3y^2: enter the power, the position and the two multipliers and get that single term without writing out the whole expansion. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- N
- 5
- Position
- 3
- Coefficient a
- 2
- Coefficient b
- -3
- Variable a
- x
- Variable b
- y
- K
- 2
- Binomial coefficient
- 10
- Coefficient
- 720
- Power a
- 3
This binomial expansion specific term calculator finds exactly one requested term of (a*x + b*y)^n without expanding the entire polynomial. Enter the non-negative exponent n and the term's one-based position, then optionally set the integer multipliers a and b and choose variable names. The result shows the binomial coefficient, the complete signed coefficient after multiplier powers are applied, each variable's exponent, and a readable term. Exact integer arithmetic keeps large coefficients reliable, while the displayed general term formula makes the method suitable for study, grading, and deterministic software workflows.
Identify the requested position in the general term formula
For the expansion of (a*x + b*y)^n, number the terms from left to right beginning with position one. The general term is T_r = C(n,r-1) * (a*x)^(n-r+1) * (b*y)^(r-1), where r is the requested position. The calculator first converts that one-based position to k = r - 1, the zero-based index commonly used in the binomial theorem.
An expansion with exponent n contains exactly n + 1 terms, so position must lie between one and n + 1. Keeping position separate from k is useful because textbook questions usually ask for the first, fourth, or seventh term, while symbolic formulas are generally written with k.
What you can do with it
Textbook specific-term exercises
Find the requested term directly, including its simplified coefficient and both exponents, without writing every preceding term of the expansion.
Automated algebra grading
Compare a student's coefficient and variable powers against exact structured output while keeping the textbook position and formula index explicit.
Lesson and worksheet generation
Generate deterministic answers for varied exponents, positions, integer multipliers, and variable names without embedding a full computer algebra system.
FAQ
Is the requested term position one-based?
Yes. Position 1 is the first term. The calculator converts it to k = position - 1 before applying the general term formula.
How many terms are in (a*x + b*y)^n?
A finite binomial expansion with non-negative integer n has n + 1 terms, so the largest valid position is n + 1.
Can this calculator find a term of (2x - 3y)^n?
The sign and multiplier powers are included in the returned complete coefficient.
How are very large exact coefficients returned?
Larger exact integers are returned as decimal strings so no digits are lost to floating-point rounding.
What is the general term formula?
The term in position r is C(n, r − 1) times (ax) to the power n − r + 1 times (by) to the power r − 1. For the third term of (2x − 3y)^5 that is 10 · (2x)^3 · (−3y)^2, or 720x^3y^2.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/algebra/binomial-specific-term \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"n":5,"position":3}'const res = await fetch("https://api.kit.forhosting.com/algebra/binomial-specific-term", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"n": 5,
"position": 3
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/algebra/binomial-specific-term",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"n": 5,
"position": 3
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/algebra/binomial-specific-term", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"n":5,"position":3}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"n":5,"position":3}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/algebra/binomial-specific-term", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"n": 5,
"position": 3
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "algebra.binomial_specific_term",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Identify the requested position in the general term formula
It then assigns power_a = n - k and power_b = k. This explains the familiar pattern: the first variable starts at power n and decreases by one from term to term, while the second starts at power zero and increases by one. For example, position three uses k = 2, making the powers n - 2 and 2. The response returns both values, removing any ambiguity about the indexing convention and making off-by-one errors easier to spot.
Understand the coefficient and variable powers
The binomial coefficient C(n,k) is only one part of a term's complete numerical coefficient when the binomial contains multipliers. A coefficient of one is omitted from a term containing variables, and a coefficient of negative one is shown as a leading minus sign. If either multiplier is zero, terms that contain a positive power of that multiplier correctly simplify to zero. All arithmetic is exact. After calculating C(n,k), the calculator multiplies it by coefficient_a raised to power_a and coefficient_b raised to power_b. Thus the requested term of (2x - 3y)^5 can have a negative coefficient even though every C(n,k) is positive: the sign comes from the power of the second multiplier. The response reports binomial_coefficient and coefficient separately so you can see the combinatorial factor and the fully simplified numerical result. It also returns power_a and power_b as integers, variable_a and variable_b as display names, and term as a compact product. Values that fit safely in JSON appear as numbers, while larger integers appear as decimal strings to preserve every digit instead of introducing floating-point rounding.
Choose inputs, validate answers, and use the result
The exponent may range from zero through the published safety limit, and position is checked against the actual number of terms for that exponent. Optional variable names let the output match notation used in a worksheet or application, but they must be short identifiers beginning with a letter so generated terms remain predictable and safe to display. On success, compare the returned powers first, then verify the binomial coefficient, and finally check the complete coefficient after multiplier powers are included. This order mirrors a clear hand solution and helps isolate mistakes. Provide n and position as JSON integers. Optional coefficient_a and coefficient_b fields default to one, which makes the input behave like the standard expansion (x + y)^n. Use negative integer multipliers for subtraction, such as coefficient_b = -1 for (x - y)^n. Invalid or missing required fields produce an invalid_input error rather than a guessed result. The free browser calculator is convenient for individual exercises, while server and SDK calls use the same deterministic contract at $0.002 per successful item. Because no network, random source, or clock participates in the calculation, identical inputs always yield identical JSON, making the capability dependable in lesson generators, assessment services, and continuous-integration checks.
Technical questions
What is the difference between binomial_coefficient and coefficient?
The binomial coefficient is C(n,k). The complete coefficient also includes coefficient_a raised to power_a and coefficient_b raised to power_b, including any resulting sign.