Question

1. a boss wants to select one group of 5 people from his 30 staff. how many different groups are possible? = enter your next step here

Explanation:

Step1: Identify the combination formula

To find the number of ways to choose \( r \) items from \( n \) items (where order doesn't matter), we use the combination formula \( _nC_r=\frac{n!}{r!(n - r)!} \). Here, \( n = 30 \) (total staff) and \( r=5 \) (people in the group).

Step2: Apply the combination formula

Substitute \( n = 30 \) and \( r = 5 \) into the combination formula: \( _{30}C_5=\frac{30!}{5!(30 - 5)!}=\frac{30!}{5!×25!} \)
We can also calculate this using the combination formula for calculators: \( _{30}C_5=\frac{30\times29\times28\times27\times26}{5\times4\times3\times2\times1} \)
Calculating the numerator: \( 30\times29\times28\times27\times26 = 30\times29 = 870; 870\times28 = 24360; 24360\times27 = 657720; 657720\times26 = 17100720 \)
Calculating the denominator: \( 5\times4\times3\times2\times1=120 \)
Then \( \frac{17100720}{120}=142506 \)

Answer:

The number of different groups possible is \( \boldsymbol{142506} \) (using the combination formula \( _{30}C_5 \) we find that there are 142506 different groups).