Committees The number of different committees of 3 students...

Committees The number of different committees of 3 students where the 3 students are selected from a class of $n$ students is given by $c(n)=\frac{1}{6} n^{3}-\frac{1}{2} n^{2}+\frac{1}{3} n$. If an art class has 10 students, how many different committees having 3 students can be selected?number of years after 2018. Determine the amount in the account in a) 2019. b) $2033 .$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Combinatorial Counting

This concept involves determining the number of distinct groups that can be formed from a larger set where the order of selection does not matter. It is frequently applied using the binomial coefficient, often expressed as n choose k, which calculates the number of ways to select k items from n.

Binomial Coefficient

A key tool in combinatorics, the binomial coefficient (denoted as C(n, k) or nCk) is defined as n!/(k!(n-k)!) and is used to count the number of ways to choose k items from n without regard to order. For selecting 3 items, it simplifies to n(n-1)(n-2)/6.

Polynomial Evaluation

In many combinatorial problems, a polynomial can represent counts that arise from binomial coefficients. Evaluating these polynomials by substituting specific values for the variable provides a method to find numerical answers to counting problems.

Compound Interest

This concept in finance refers to the process by which interest accrues on both the initial principal and the accumulated interest from previous periods, resulting in exponential growth over time. Calculations typically use exponential functions to determine the future value of an investment based on the principal, interest rate, and time elapsed.

Transcript

Please give Ace some feedback