Winning the jackpot in a particular lottery requires that you select the correct four numbers between 1 and 48 and, in a separate drawing, you must also select the correct single number between 1 and 48 . Find the probability of winning the jackpot. The probability of winning the jackpot is (Type an integer or simplified fraction.)

To find the probability of winning the jackpot in this lottery, we need to calculate the total number of possible outcomes for both the selection of the four numbers and the selection of the single number. 1. **Selecting the four numbers:** - You need to choose 4 numbers from a set of 48. The order in which you select the numbers does not matter, so we use combinations. - The number of ways to choose 4 numbers from 48 is given by the combination formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] - Here, \( n = 48 \) and \( k = 4 \): \[ \binom{48}{4} = \frac{48!}{4!(48-4)!} = \frac{48 \times 47 \times 46 \times 45}{4 \times 3 \times 2 \times 1} = \frac{48 \times 47 \times 46 \times 45}{24} \] - Calculating this step-by-step: - \( 48 \times 47 = 2256 \) - \( 2256 \times 46 = 103776 \) - \( 103776 \times 45 = 4669920 \) - Now divide by 24: \[ \frac{4669920}{24} = 194580 \] 2. **Selecting the single number:** - There are 48 possible choices for the single number. 3. **Total number of outcomes:** - The total number of outcomes for the lottery is the product of the number of ways to choose the four numbers and the number of ways to choose the single number: \[ \text{Total outcomes} = \binom{48}{4} \times 48 = 194580 \times 48 \] - Calculating this: \[ 194580 \times 48 = 9339840 \] 4. **Probability of winning the jackpot:** - There is only one winning combination (the correct four numbers and the correct single number). - Therefore, the probability of winning the jackpot is: \[ P(\text{winning}) = \frac{1}{\text{Total outcomes}} = \frac{1}{9339840} \] Thus, the probability of winning the jackpot is: \[ \frac{1}{9339840} \]