91Ó°ÊÓ

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(x-3 y)^{5}$$

Evaluate each expression. $$\frac{_{5} C_{1} \cdot_{7} C_{2}}{_{12} C_{3}}$$

Write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for \(a_{n}\) to find \(a_{20},\) the 20 th term of the sequence. $$a_{1}=6, d=3$$

Evaluate each factorial expression. $$\frac{(2 n+1) !}{(2 n) !}$$

Use the formula for the sum of the first \(n\) terms of a geometric sequence to solve. Find the sum of the first 11 terms of the geometric sequence: \(4,-12,36,-108, \dots\)

Find each indicated sum. $$\sum_{i=1}^{6} 5 i$$

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(2 a+b)^{6}$$

Write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for \(a_{n}\) to find \(a_{20},\) the 20 th term of the sequence. $$a_{1}=-20, d=-4$$

Use the formula for the sum of the first \(n\) terms of a geometric sequence to solve. Find the sum of the first 14 terms of the geometric sequence: \(-\frac{3}{2}, 3,-6,12, \ldots\)

Mega Millions is a multi-state lottery played in most U.S. states. As of this writing, the top cash prize was \(\$ 656\) million, going to three lucky winners in three states. Players pick five different numbers from 1 to 56 and one number from 1 to \(46 .\) Use this information to solve Exercises \(27-30 .\) Express all probabilities as fractions. A player wins a minimum award of \(\$ 150\) by correctly matching three numbers drawn from white balls (1 through 56) and matching the number on the gold Mega Ball" ( 1 through 46 ). What is the probability of winning this consolation prize?