Question

a class has 34 students. in how many different ways can four students form a group for an activity? (assume the order of the students is not important.)
there are different ways that the four students can form a group for an activity.
(type a whole number.)

Explanation:

Step1: Identify the combination formula

Since the order of students is not important, use the combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 34\) (total number of students) and \(r=4\) (number of students in a group).

Step2: Calculate factorial values

First, calculate \(n!=34! = 34\times33\times32\times31\times30!\), \(r!=4! = 4\times3\times2\times1=24\), and \((n - r)!=(34 - 4)!=30!\).
Then, \(C(34,4)=\frac{34!}{4!(34 - 4)!}=\frac{34\times33\times32\times31\times30!}{24\times30!}\).

Step3: Simplify the expression

Cancel out the \(30!\) terms. Then \(\frac{34\times33\times32\times31}{24}\).
\(34\times33\times32\times31=(30 + 4)\times33\times32\times31=(30\times33+4\times33)\times32\times31=(990+132)\times32\times31 = 1122\times32\times31=(1100+22)\times32\times31=(1100\times32+22\times32)\times31=(35200+704)\times31=35904\times31 = 1113024\).
And \(\frac{1113024}{24}=46376\).

Answer:

\(46376\)