A lottery game has players selecting 4 numbers at random from the numbers \( 1-45 \). The state picks 20 winning numbers. If the player's 4 numbers are selected by the state, the player wins. What is the probability of winning? How many ways can the 4 numbers be chosen from the original 45 numbers, regardless of whether they are in the winning group or the losing group? \( \square \) ways (Type a whole number.) The probability of winning is \( \square \). (Round to four decimal places as needed.)

Let the player’s chosen 4 numbers be any 4 numbers out of 45 and the state’s winning set be 20 numbers out of 45. The player wins if all his 4 numbers are among the state’s 20 winning numbers. **Step 1. Compute the total number of ways to choose 4 numbers from 45.** The number of ways is given by the combination \[ \binom{45}{4}=\frac{45\times44\times43\times42}{4\times3\times2\times1}. \] We simplify by canceling factors: - First, note that \(4\times3\times2\times1=24\). - Divide \(44\) by \(4\) to get \(11\) and \(42\) by \(2\) to get \(21\): \[ \frac{45\times11\times43\times21}{3}. \] - Now, divide \(45\) by \(3\) to get \(15\): \[ 15\times11\times43\times21. \] Now compute: \[ 15\times11=165, \] \[ 165\times43=7095, \] \[ 7095\times21=148995. \] Thus, there are \[ 148995 \text{ ways.} \] **Step 2. Compute the probability of winning.** A player wins if his 4 numbers are among the 20 winning numbers. The number of combinations for choosing 4 numbers from the 20 winning numbers is: \[ \binom{20}{4}=\frac{20\times19\times18\times17}{4\times3\times2\times1}. \] Again, the denominator is \(24\): - Compute the numerator: \(20\times19=380\); then \(380\times18=6840\); finally \(6840\times17=116280\). - Now divide: \[ 116280\div24=4845. \] Thus, there are \(4845\) ways for the player’s 4 chosen numbers to be a subset of the 20 winning numbers. The probability is the ratio: \[ \text{Probability}=\frac{\binom{20}{4}}{\binom{45}{4}}=\frac{4845}{148995}. \] Dividing, we get approximately: \[ \frac{4845}{148995}\approx0.0325. \] **Final Answers:** 148995 ways 0.0325