14. a teacher wants to select one group of 5 people from his 30 students. how many different groups are…

Explanation:

Step1: Identify the problem type

This is a combination problem where we need to find the number of ways to choose 5 students from 30, which is calculated using the combination formula ( C(n, k) = \frac{n!}{k!(n - k)!} ), where ( n = 30 ) and ( k = 5 ).

Step2: Apply the combination formula

First, calculate the factorial values:

• ( n! = 30! = 30 \times 29 \times 28 \times 27 \times 26 \times 25! )
• ( k! = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 )
• ( (n - k)! = (30 - 5)! = 25! )

Now, substitute these into the combination formula: [ C(30, 5) = \frac{30!}{5!(30 - 5)!} = \frac{30 \times 29 \times 28 \times 27 \times 26 \times 25!}{120 \times 25!} ] The ( 25! ) terms cancel out, leaving: [ C(30, 5) = \frac{30 \times 29 \times 28 \times 27 \times 26}{120} ]

Step3: Calculate the numerator

Calculate ( 30 \times 29 \times 28 \times 27 \times 26 ): [ 30 \times 29 = 870 \ 870 \times 28 = 24360 \ 24360 \times 27 = 657720 \ 657720 \times 26 = 17100720 ]

Step4: Divide by the denominator

Now, divide ( 17100720 ) by ( 120 ): [ \frac{17100720}{120} = 142506 ] Wait, no, wait, let's recalculate the multiplication correctly:

Wait, actually, let's compute step by step:

( 30 \times 29 = 870 )

( 870 \times 28 = 870 \times 20 + 870 \times 8 = 17400 + 6960 = 24360 )

( 24360 \times 27 = 24360 \times 20 + 24360 \times 7 = 487200 + 170520 = 657720 )

( 657720 \times 26 = 657720 \times 20 + 657720 \times 6 = 13154400 + 3946320 = 17100720 )

Now divide by 120: ( 17100720 \div 120 = 142506 )? Wait, no, wait, let's check the calculation again. Wait, maybe I made a mistake in multiplication. Wait, let's use a better approach:

[ \frac{30 \times 29 \times 28 \times 27 \times 26}{5 \times 4 \times 3 \times 2 \times 1} ]

Simplify step by step:

30 divided by 5 is 6.

28 divided by 4 is 7.

27 divided by 3 is 9.

26 divided by 2 is 13.

So now we have:

6 (from 30/5) × 29 × 7 (from 28/4) × 9 (from 27/3) × 13 (from 26/2)

Now multiply these:

6 × 29 = 174

174 × 7 = 1218

1218 × 9 = 10962

10962 × 13 = 142506

Wait, but the options include 142559, 142615, 142391, 142506. Wait, but according to the calculation, it's 142506, which is option D. Wait, but let me check the combination formula again. Wait, maybe I made a mistake in the formula? No, combination is used when the order doesn't matter, which is the case here (selecting a group, order within the group doesn't matter). So the calculation should be correct.

Wait, let's verify with a calculator:

( C(30,5) = \frac{30!}{5!25!} = \frac{30×29×28×27×26}{5×4×3×2×1} = \frac{17100720}{120} = 142506 ). So the correct answer is D.

Answer:

D. 142 506