Factorials, combinations and binomial coefficients
| English | 中文 | Pinyin |
|---|---|---|
| combination/ˌkɒmbɪˈneɪʃn/ | 组合 | zǔ hé |
A binomial coefficient counts which factors contribute an x term. Why is the x² coefficient not just the middle number in Pascal’s triangle?
- A binomial coefficient counts which factors contribute an x term. Why is the x² coefficient not just the middle number in Pascal’s triangle?
- This lesson studies combination 组合: A selection in which order does not distinguish different choices.
Choose the mathematical structure
- For a nonnegative integer n, n!=n(n−1)…1 and 0!=1. For 0≤r≤n, nCr=n!/[r!(n−r)!] counts unordered selections. In (a+bx)^n with positive integer n, the x^r coefficient is nCr × a^(n−r) × b^r. The term number is r+1 when the expansion is written in ascending powers of x.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines combination?
A selection in which order does not distinguish different choices.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
4!=24 and 4C2=24/(2×2)=6. The coefficients for n=4 are 1,4,6,4,1. In (2+3x)^4, the x² term is 4C2×2²×(3x)²=216x²; the constant is 2⁴=16 and the x term is 4×2³×3x=96x. For four independent trials each with success probability 1/3, exactly two successes have probability 4C2×(1/3)²×(2/3)²=8/27. The coefficient 6 counts the possible success positions; each arrangement has the same probability.
Factorials, combinations and binomial coefficients
For a nonnegative integer n, n!=n(n−1)…1 and 0!=1
Connect a factored expression to its graph or expansion coefficients.
Evaluate 4!.
4!=4×3×2×1=24.
Test a tempting shortcut
- Do not omit the powers of a or b when using a combination coefficient. The x² term is the third term, not the second. The probability expression needs a fixed number of trials, independent binary outcomes and the same success probability; a changing probability invalidates this simple model.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The x² coefficient of (a+bx)^n is always just nC2. This claim is false. Explain which definition or assumption it violates.
Evaluate 4C2.
4C2=4!/(2!2!)=24/4=6.
The x² coefficient of (a+bx)^n is always just nC2.
Do not omit the powers of a or b when using a combination coefficient. The x² term is the third term, not the second. The probability expression needs a fixed number of trials, independent binary outcomes and the same success probability; a changing probability invalidates this simple model.
Interpret a new situation
- Use symmetry nCr=nC(n−r) and cancel factorial factors before calculating. To find the x³ coefficient in (2+3x)^4, use 4C3×2×3³=216. The finite polynomial formula here is for positive integer n; a rational exponent instead gives the separate general binomial series with its stated validity interval.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the coefficient of x² in (2+3x)^4.
The coefficient is 6×2²×3²=216.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · D. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A selection in which order does not distinguish different choices. Choose the relationship, show the method, check its assumptions and interpret the result.