Question

1. a teacher wants to select one group of 5 people from his 30 students. how many different groups are possible? 142 559 a 142 615 b 142 391 c 142 506 d

Explanation:

Step1: Identify the problem type

This is a combination problem where we want to choose 5 students out of 30, and the order of selection doesn't matter. The formula for combinations is $C(n, k)=\frac{n!}{k!(n - k)!}$, where $n = 30$ and $k=5$.

Step2: Calculate the factorial values

First, calculate the factorials:
- $n!=30! = 30\times29\times28\times27\times26\times25!$
- $k!=5! = 5\times4\times3\times2\times1 = 120$
- $(n - k)!=(30 - 5)!=25!$

Step3: Substitute into the combination formula

Substitute these values into the combination formula:
\[ \begin{align*} C(30, 5)&=\frac{30!}{5!(30 - 5)!}\\ &=\frac{30\times29\times28\times27\times26\times25!}{120\times25!}\\ &=\frac{30\times29\times28\times27\times26}{120} \end{align*} \]

Step4: Perform the multiplication and division

First, calculate the numerator: $30\times29 = 870$, $870\times28 = 24360$, $24360\times27 = 657720$, $657720\times26 = 17100720$
Then divide by the denominator: $\frac{17100720}{120}=142506$? Wait, no, wait, let's recalculate:

Wait, $30\times29\times28\times27\times26$:

$30\times29 = 870$

$870\times28 = 870\times(20 + 8)=870\times20+870\times8 = 17400+6960 = 24360$

$24360\times27$: $24360\times20 = 487200$, $24360\times7 = 170520$, so $487200+170520 = 657720$

$657720\times26$: $657720\times20 = 13154400$, $657720\times6 = 3946320$, so $13154400+3946320 = 17100720$

Now divide by 120: $17100720\div120$. Let's divide numerator and denominator by 10 first: $1710072\div12$. Then divide 1710072 by 12: $1710072\div12 = 142506$? Wait, but that's option D? Wait, no, I must have made a mistake. Wait, let's recalculate the combination:

Wait, the correct formula for $C(n,k)=\frac{n(n - 1)(n - 2)(n - 3)(n - 4)}{k(k - 1)(k - 2)(k - 3)(k - 4)}$ for $k = 5$:

So $C(30,5)=\frac{30\times29\times28\times27\times26}{5\times4\times3\times2\times1}$

Calculate numerator: $30\times29 = 870$, $870\times28 = 24360$, $24360\times27 = 657720$, $657720\times26 = 17100720$

Denominator: $5\times4\times3\times2\times1=120$

Now divide $17100720\div120$:

$17100720\div120 = 17100720\div(10\times12)= (17100720\div10)\div12 = 1710072\div12$

$1710072\div12$: $12\times142506 = 1710072$? Wait, but let's check with another approach.

Alternatively, use the formula $C(n,k)=\frac{n!}{k!(n - k)!}$, and we can calculate step by step:

$C(30,5)=\frac{30\times29\times28\times27\times26}{5\times4\times3\times2\times1}$

Simplify the fraction before multiplying:

30 and 5: $30\div5 = 6$

28 and 4: $28\div4 = 7$

27 and 3: $27\div3 = 9$

26 and 2: $26\div2 = 13$

So now we have:

$6\times29\times7\times9\times13$

Calculate this:

$6\times29 = 174$

$174\times7 = 1218$

$1218\times9 = 10962$

$10962\times13$: $10962\times10 = 109620$, $10962\times3 = 32886$, so $109620+32886 = 142506$? Wait, but that's option D? But wait, maybe I made a mistake in the problem. Wait, no, wait, let's check with a calculator for $C(30,5)$:

The formula for combinations: $C(n,k)=\frac{n!}{k!(n - k)!}$, so $C(30,5)=\frac{30!}{5!25!}=\frac{30\times29\times28\times27\times26}{5\times4\times3\times2\times1}=142506$? Wait, but the options have D as 142506? Wait, but let's check again. Wait, maybe I miscalculated. Wait, 30 choose 5:

$30\times29 = 870$

$870\times28 = 24360$

$24360\times27 = 657720$

$657720\times26 = 17100720$

Divide by 120: $17100720\div120 = 142506$. So the answer should be D. Wait, but let's check the options again:

A: 142559, B:142615, C:142391, D:142506. So according to the calculation, the correct answer is D.

Wait, but maybe I made a mistake in the simplification. Wait, let's do it again:

$C(30,5)=\frac{30!}{5!25!}=\frac{30\times29\times28\times27\times26}{5\times4\times3\times2\times1}$

Simplify numerator and denominator:

30/5 = 6, 28/4 = 7, 27/3 = 9, 26/2 = 13. So now we have 6×29×7×9×13.

6×29 = 174

174×7 = 1218

1218×9 = 10962

10962×13: 10962×10 = 109620, 10962×3 = 32886, sum is 109620 + 32886 = 142506. Yes, that's correct. So the answer is D.

Answer:

D. 142506