Problem 56
Question
To win in the New York State lottery, one must correctly select 6 numbers from 59 numbers. The order in which the selection is made does not matter. How many different selections are possible?
Step-by-Step Solution
Verified Answer
The total number of different selections possible is the value obtained in Step 4.
1Step 1: Understanding Combinations
Combinations can be calculated using the formula for combinations which is \( C(n, k) = \frac{n!}{k!(n-k)!} \), where n is the total number of items, k is the number of items to choose, and '!' indicates factorial.
2Step 2: Applying the combinations formula
Given that there are 59 numbers (n=59) and we need to select 6 numbers (k=6), we can simply substitute these values into the combinations formula. So, the calculation would be \( C(59, 6) = \frac{59!}{6!(59-6)!} \).
3Step 3: Calculating Factorials
The factorial function can be computed as the product of all positive integers up to the designated number. \( 59! = 59*58*57*...*1 \), \( 6! = 6*5*4*3*2*1 \), and \( (59-6)! = 53*52*51*...*1 \). These values can be computed with a calculator.
4Step 4: Performing the Division
After calculating the factorials, divide the numerator by the denominator to get the total number of combinations.
Other exercises in this chapter
Problem 56
Find the middle term in the expansion of \(\left(\frac{1}{x}-x^{2}\right)^{12}\)
View solution Problem 56
express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. $$ 6+8+10+12+\cdots+32 $$
View solution Problem 57
Explaining the Concepts Give an example of an event whose probability must be determined empirically rather than theoretically.
View solution Problem 57
In Exercises \(57-62,\) let $$ \begin{array}{l} {\left\\{a_{n}\right\\}=-5,10,-20,40, \ldots} \\ {\left\\{b_{n}\right\\}=10,-5,-20,-35, \ldots} \end{array} $$ a
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