Pascal's Triangle and Binomial Coefficients

Explore combination count, pascal recurrence, binomial coefficients.

Lesson 4 of 53 parts Practice Flashcards

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Combination Count

Objective: Explain combination count and use it to solve or interpret a problem.

C(n,k)=n!/[k!(n−k)!] counts unordered selections from n distinct objects, with 0≤k≤n. For arrangements or assigned roles, use a permutation count.

Worked examples

Try itPascal's Triangle and Binomial Coefficients — practice 1.1: How many ways can 2 students be selected from 7 students?Show answer

Step 1: Order does not matter, so use C(7,2). Step 2: Compute (7)(6)/2. Step 3: The value is 21.

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Combination Count

Objective: Explain combination count and use it to solve or interpret a problem.

C(n,k)=n!/[k!(n−k)!] counts unordered selections from n distinct objects, with 0≤k≤n. For arrangements or assigned roles, use a permutation count.

Worked examples

Try itPascal's Triangle and Binomial Coefficients — practice 1.1: How many ways can 2 students be selected from 7 students?Show answer

Step 1: Order does not matter, so use C(7,2). Step 2: Compute (7)(6)/2. Step 3: The value is 21.

Pascal Recurrence

Objective: Explain pascal recurrence and use it to solve or interpret a problem.

Each interior entry of Pascal triangle equals the sum of two entries directly above; C(n,k)=C(n−1,k−1)+C(n−1,k). They are one.

Worked examples

Try itPascal's Triangle and Binomial Coefficients — practice 2.1: In Pascal’s triangle, find C(10,1).Show answer

Step 1: C(n,1) counts one choice from n objects. Step 2: Substitute n=10. Step 3: The coefficient is 10.

Binomial Coefficients

Objective: Explain binomial coefficients and use it to solve or interpret a problem.

The coefficients of (u+v)^n follow row n of Pascal triangle when n is a nonnegative integer; count exponents carefully. Each factor contributes exactly one u or v.

Worked examples

Try itPascal's Triangle and Binomial Coefficients — practice 3.1: Find the x coefficient of (x+3)^11 for positive integer c.Show answer

Step 1: Choose one x from the 11 factors. Step 2: The other 10 factors contribute 3^10. Step 3: Multiply 11·59049=649539.

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