question 2 of 10\nat the end of a weeklong seminar, the presenter decides to give away signed copies of his…

question 2 of 10\nat the end of a weeklong seminar, the presenter decides to give away signed copies of his book to 4 randomly selected people in the audience.\nhow many different ways can this be done if 30 people are present at the seminar?

question 2 of 10\nat the end of a weeklong seminar, the presenter decides to give away signed copies of his book to 4 randomly selected people in the audience.\nhow many different ways can this be done if 30 people are present at the seminar?

Answer

Explanation:

Step1: Identify the combination formula

The number of combinations of (n) objects taken (r) at a time is given by (C(n,r)=\frac{n!}{r!(n - r)!}), where (n = 30) (total number of people) and (r=4) (number of people to be selected).

Step2: Calculate factorial values

First, calculate (n!=30!), (r!=4!) and ((n - r)!=26!). Then (C(30,4)=\frac{30!}{4!(30 - 4)!}=\frac{30!}{4!×26!}). Since (n!=n\times(n - 1)\times\cdots\times(n - r+1)\times(n - r)!), we can simplify (\frac{30!}{26!}=30\times29\times28\times27). So (C(30,4)=\frac{30\times29\times28\times27}{4!}). And (4!=4\times3\times2\times1 = 24).

Step3: Compute the result

(C(30,4)=\frac{30\times29\times28\times27}{24}) [ \begin{align*} \frac{30\times29\times28\times27}{24}&=\frac{30\times29\times7\times27}{6}\ &= 5\times29\times7\times27\ &=145\times189\ &=27405 \end{align*} ]

Answer:

(27405)