The Mathematics Performance Review
Mr. Umar recently compiled the test scores of a small study group of 9 students. The scores, marked out of 20, were: 10, 12, 12, 14, 15, 16, 18, 18, and 18.
To analyze the performance, he first wanted to find the measures of central tendency and spread. By looking at the highest and lowest scores, he calculated the range of the test scores to be (1) points. Next, he identified the score that appeared the most frequently, which means the mode of this dataset is (2) . To find the middle ground, he arranged the 9 scores in ascending order and located the middle value, revealing that the median score is (3) . Finally, he added all 9 scores together to get a total sum of (4) . By dividing this total sum by the number of students, he calculated the precise mean score of the group to be (5) .
Mr. Umar decided to reward the students based on these results. If he randomly picks one student's test script from the pile of 9, the probability that the student scored exactly the mode is (6) , while the probability of picking a script with a score strictly less than the median is (7) . Furthermore, the probability that a randomly chosen student scored at least 15 marks is (8) . If he picks a script at random, the probability that the score is a prime number is (9) , and the probability that the score is a perfect square is (10) .
Phase 2: Organizing the Students
After analyzing the scores, Mr. Umar brought the 3 students who scored 18 and the 2 students who scored 12 into a separate room for a special project. He wanted to see how many ways he could arrange them.
If he wants to arrange all 5 of these specific students in a straight line, the total number of unique permutations possible is (11) . However, because the three students who scored 18 are identical triplets, if he treats the triplets as indistinguishable from each other, the number of distinct linear arrangements drops to (12) . If he goes back to treating all 5 students as unique individuals but insists that the 2 students who scored 12 must sit next to each other, the number of ways to arrange them in a line is (13) . If he arranges all 5 unique students around a circular discussion table instead of a straight line, the number of distinct arrangements is (14) .
From the total group of 9 students, Mr. Umar needs to select leadership roles. If he wants to choose a Class Captain, a Deputy Captain, and a Timekeeper, where no student can hold more than one role, the number of ways he can fill these 3 distinct positions is (15) . If he then takes the 4 top-performing students and wants to arrange them on a display board photo featuring a row of just 2 chairs, the number of ways to seat them is (16)
Phase 3: Committee Selection and Probabilities
Instead of assigning specific roles, Mr. Umar decides to form general committees. He needs to choose a committee of 4 students from the total group of 9. The number of unique combinations for this committee is (17) . If he instead decides to form a smaller committee of 7 students from the 9, the number of possible combinations is (18)
The group of 9 students consists of 5 boys and 4 girls. Mr. Umar wants to form a new committee of 3 students. The total number of ways to choose any 3 students from the 9 is (19) . If he restricts the committee so that it must contain exactly 2 boys and 1 girl, the number of ways to form it is (20) . The probability that a randomly selected 3-member committee consists of exactly 2 boys and 1 girl is (21) . If he forms another 3-member committee, the number of ways to select a group containing only girls is (22) , which means the probability of choosing a committee of all girls is (23)
Phase 4: Independent and Dependent Events
To wrap up the session, Mr. Umar introduces a game with two fair, six-sided dice. If a student rolls both dice simultaneously, the total number of possible outcomes in the sample space is (24) . The probability of rolling a total sum of exactly 7 is (25) , while the probability of rolling a "double" (the same number on both dice) is (26) . The probability of rolling a sum that is less than or equal to 4 is (27)
Finally, he places 10 red pens and 6 blue pens into a drawer. If a student draws one pen at random, notes its color, replaces it, and draws a second pen, the probability that both pens are red is (28) . If the student draws two pens one after the other without replacement, the total number of ways to choose 2 pens out of 16 is (29) , and the probability that both selected pens are blue is (30)
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