Binomial Theorem and Applications

Explore general term, selected coefficient, applied patterns.

Lesson 5 of 53 parts Practice Flashcards

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General Term

Objective: Explain general term and use it to solve or interpret a problem.

The term C(n,k)u^(n−k)v^k appears in (u+v)^n for k=0,…,n. Identify the requested power before expanding everything. It counts ways to choose the k factors contributing v.

Worked examples

Try itBinomial Theorem and Applications — practice 1.1: Find the x coefficient of (x+2)^5 for positive integer c.Show answer

Step 1: Choose one x from the 5 factors. Step 2: The other 4 factors contribute 2^4. Step 3: Multiply 5·16=80.

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General Term

Objective: Explain general term and use it to solve or interpret a problem.

The term C(n,k)u^(n−k)v^k appears in (u+v)^n for k=0,…,n. Identify the requested power before expanding everything. It counts ways to choose the k factors contributing v.

Worked examples

Try itBinomial Theorem and Applications — practice 1.1: Find the x coefficient of (x+2)^5 for positive integer c.Show answer

Step 1: Choose one x from the 5 factors. Step 2: The other 4 factors contribute 2^4. Step 3: Multiply 5·16=80.

Selected Coefficient

Objective: Explain selected coefficient and use it to solve or interpret a problem.

To find one coefficient, match the variable power to k and calculate that term; a full expansion may be unnecessary. Its power contributes an additional numeric factor.

Worked examples

Try itBinomial Theorem and Applications — practice 2.1: Find the x coefficient of (x+3)^8 for positive integer c.Show answer

Step 1: Choose one x from the 8 factors. Step 2: The other 7 factors contribute 3^7. Step 3: Multiply 8·2187=17496.

Applied Patterns

Objective: Explain applied patterns and use it to solve or interpret a problem.

Binomial coefficients also count combinations, while an algebraic expansion includes powers and signs. Interpret which part a context asks for. No, event probabilities also need outcome probabilities and assumptions.

Worked examples

Try itBinomial Theorem and Applications — practice 3.1: How many ways can 2 students be selected from 13 students?Show answer

Step 1: Order does not matter, so use C(13,2). Step 2: Compute (13)(12)/2. Step 3: The value is 78.

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