2. a boss wants to select one group of 5 people from his 30 staff. how many different groups are possible? =…

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 to select).

Step2: Substitute values into the formula

Substitute ( n = 30 ) and ( r = 5 ) into ( _{30}C_5 ). So we have ( _{30}C_5=\frac{30!}{5!(30 - 5)!}=\frac{30!}{5!×25!} ). We know that ( n!=n\times(n - 1)\times\cdots\times1 ), so ( 30! = 30\times29\times28\times27\times26\times25! ). Substituting this into the formula: ( _{30}C_5=\frac{30\times29\times28\times27\times26\times25!}{5!×25!} ) The ( 25! ) terms cancel out. And ( 5!=5\times4\times3\times2\times1 = 120 ). So ( _{30}C_5=\frac{30\times29\times28\times27\times26}{120} ) Calculate the numerator: ( 30\times29 = 870 ), ( 870\times28=24360 ), ( 24360\times27 = 657720 ), ( 657720\times26=17100720 ) Then divide by 120: ( \frac{17100720}{120}=142506 )

Answer:

( 142506 )