Question

in a certain lottery, 5 numbers between 1 and 13 inclusive are drawn. these are the winning numbers. how many different selections are possible? assume that the order in which the numbers are drawn is not important.
a. 1287
c. 371,293
b. 154,440
d. 120
please select the best answer from the choices provided
○ a
○ b
○ c
○ d

Explanation:

Step1: Identify the combination formula

The formula for combinations is \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n\) is the total number of items, and \(r\) is the number of items to be chosen. Here, \(n = 13\) (numbers from 1 to 13) and \(r=5\) (5 numbers are drawn).

Step2: Calculate factorial values

First, calculate \(n!=13! = 13\times12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1\), \(r!=5! = 5\times4\times3\times2\times1\), and \((n - r)!=(13 - 5)!=8! = 8\times7\times6\times5\times4\times3\times2\times1\). Then \(C(13,5)=\frac{13!}{5!(13 - 5)!}=\frac{13\times12\times11\times10\times9\times8!}{5!\times8!}\).

Step3: Simplify the expression

Cancel out the \(8!\) terms. So \(C(13,5)=\frac{13\times12\times11\times10\times9}{5\times4\times3\times2\times1}\).

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Answer:

A. 1287