answer:# Solution: Part 1: Given data: - Total graduate students majoring in physics: 200 - Students who like employment in patent agency: 140 - Students who dislike employment in patent agency: 60 First, we calculate the probability of a student liking employment in the patent agency: [P(text{Likes}) = frac{text{Number of students who like}}{text{Total number of students}} = frac{140}{200} = frac{7}{10}] When 3 students are selected, we want to find the probability of at least 2 liking employment in the patent agency. Let X be the number of students who like employment in the patent agency. We calculate P(X geq 2) as follows: [P(X geq 2) = P(X = 2) + P(X = 3)] [P(X = 2) = {C}_{3}^{2} times left(frac{7}{10}right)^{2} times frac{3}{10}] [P(X = 3) = {C}_{3}^{3} times left(frac{7}{10}right)^{3}] [P(X geq 2) = 3 times left(frac{7}{10}right)^{2} times frac{3}{10} + left(frac{7}{10}right)^{3}] [P(X geq 2) = frac{3 times 49 times 3}{1000} + frac{343}{1000}] [P(X geq 2) = frac{441}{1000} + frac{343}{1000}] [P(X geq 2) = frac{784}{1000} = frac{98}{125}] Therefore, the probability of selecting at least 2 graduate students who like employment in the patent agency from the graduate students majoring in physics when 3 students are selected is boxed{frac{98}{125}}. Part 2: Given data for the chi^{2} test: - Total students (n): 200 - Male students liking patent agency (a): 60 - Male students disliking patent agency (b): 40 - Female students liking patent agency (c): 80 - Female students disliking patent agency (d): 20 The null hypothesis H_{0} states that there is no association between the intention to work in the patent agency and gender. We calculate the chi^{2} statistic as follows: [chi^{2} = frac{n(ad-bc)^{2}}{(a+b)(c+d)(a+c)(b+d)}] [chi^{2} = frac{200(60 times 20 - 40 times 80)^{2}}{(60+40)(80+20)(60+80)(40+20)}] [chi^{2} = frac{200(1200 - 3200)^{2}}{100 times 100 times 140 times 60}] [chi^{2} = frac{200(2000)^{2}}{8400000}] [chi^{2} = frac{8000000}{8400000}] [chi^{2} = frac{200}{21} approx 9.524] Comparing chi^{2} to the critical value from the table for alpha = 0.005, we have: [9.524 > 7.879] Since the calculated chi^{2} value is greater than the critical value, we reject the null hypothesis H_{0}. This means there is an association between the intention of graduate students majoring in physics to work in the patent agency and gender, with a probability of making this inference error being less than 0.005. Therefore, the conclusion is that the intention of graduate students majoring in physics to work in the patent agency is related to gender, and the probability of making an error in this inference is less than 0.005. The final answer is boxed{text{Yes, there is a relation}}.

answer:The inequality frac{1}{|m|} geq frac{1}{5} simplifies to |m| leq 5 due to the positive nature of |m| and the properties of inequality when dealing with reciprocals. This inequality is satisfied for -5 leq m leq 5. Among these, the even integers are -4, -2, 0, 2, 4. Since m neq 0, we exclude 0 from this list. Thus, the even integers that satisfy the given condition are -4, -2, 2, 4. Counting these, we have a total of 4 such integers. Therefore, the final answer is boxed{4}.

answer:Using the property of binomial coefficients, we know that: - binom{n}{n} = 1 for any non-negative integer n. - Therefore, binom{150}{150} = 1. This can be shown using the binomial coefficient formula: [ binom{150}{150} = frac{150!}{150! times (150-150)!} = frac{150!}{150! times 0!} ] Since 0! = 1, this simplifies to: [ binom{150}{150} = frac{150!}{150! times 1} = frac{150!}{150!} = 1 ] Thus, the answer is boxed{1}.

answer:First, let's calculate how many votes James received with 0.5 percent of the 2000 votes cast: 0.5% of 2000 votes = (0.5/100) * 2000 = 0.005 * 2000 = 10 votes James received 10 votes. To win the election, a candidate needed to receive more than 50 percent of the votes. Let's calculate what 50 percent of the votes would be: 50% of 2000 votes = (50/100) * 2000 = 0.5 * 2000 = 1000 votes However, to win, a candidate needs more than 50 percent, so James would need at least 1001 votes to win the election. Now, let's calculate how many additional votes James would have needed to reach this number: Additional votes needed = 1001 votes - 10 votes (that James already has) = 991 votes James would have needed an additional boxed{991} votes to win the election.