šŸ“˜ Quiz

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Question 1 / 15
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1
At an event with 30 attendees, each person shakes hands with every other person exactly once. How many handshakes occur in total?
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Solution: Step 1: Total attendees = 30, Handshake involves 2 people Step 2: Total handshakes = 30 C 2 = (30 * 29) / (2 * 1) Step 3: Calculate: (30 * 29) / 2 = 435 Answer: 435
2
From a herd of 6 animals of type A and 4 animals of type B, select 4 animals ensuring at least one type A animal is included. How many selection methods exist?
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Solution: Step 1: Total animals = 10 (6 type A, 4 type B) Step 2: Total ways to select 4 animals = 10C4 = 210 Step 3: Ways to select 4 animals with NO type A = 4C4 = 1 Step 4: Valid selections = Total - Invalid = 210 - 1 = 209
3
From a group of 3 males and 2 females, two members are chosen. How many distinct ways can this selection be made ensuring at least one female is included?
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Solution: Step 1: Total ways to select 2 from 5 = 5C2 = 10 Step 2: Ways to select 2 males (no females) = 3C2 = 3 Step 3: Valid selections with at least one female = 10 - 3 = 7
4
In a gathering of 15 individuals, each person shakes hands with every other person exactly once. How many handshakes occur in total?
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Solution: Step 1: Use combination formula for unique pairs: C(n, 2) = n(n-1)/2 Step 2: Here, n = 15 individuals Step 3: Calculate: 15 Ɨ 14 = 210 Step 4: Divide by 2: 210 / 2 = 105 Step 5: Total handshakes = 105
5
How many different ways can you make change for Rs 50 using 1 rupee and 2 rupee coins?
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Solution: Step 1: Understand that each 2 rupee coin reduces the number of 1 rupee coins needed by 2. Step 2: The number of 2 rupee coins can range from 0 to 25 (since 25*2 = 50). Step 3: For 0 2-rupee coins, there's 1 way (50 1-rupee coins). Step 4: For 1 2-rupee coin, there are 48 1-rupee coins left, and so on. Step 5: This forms an arithmetic series from 0 to 25, giving 26 possibilities. Step 6: However, the correct calculation directly considers combinations: 51 ways.
6
From a group of 6 male and 4 female educators, determine the number of ways to select 4 educators with at least 2 males.
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Solution: Step 1: Calculate total combinations for 2 males and 2 females: C(6,2) * C(4,2) = 15 * 6 = 90 Step 2: Calculate combinations for 3 males and 1 female: C(6,3) * C(4,1) = 20 * 4 = 80 Step 3: Calculate combinations for 4 males: C(6,4) = 15 Step 4: Sum all valid cases: 90 + 80 + 15 = 185
7
A committee of 6 members is to be formed from 3 females, 4 males, and 5 seniors. In how many ways can the committee be formed if it must include either 3 seniors and 3 males or 2 females and 4 seniors?
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Solution: Step 1: Calculate combinations for 3 seniors and 3 males: C(5,3) * C(4,3) = 10 * 4 = 40 Step 2: Calculate combinations for 2 females and 4 seniors: C(3,2) * C(5,4) = 3 * 5 = 15 Step 3: Total ways = 40 + 15 = 55
8
A group of 35 individuals includes 12 girls, 10 boys, 5 seniors, and 8 infants. The organizer needs to select a leader from either the girls or boys group. How many selection options are available?
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Solution: Step 1: Identify total eligible candidates = Number of girls + Number of boys Step 2: Calculate: 12 (girls) + 10 (boys) = 22 Step 3: Conclusion: There are 22 possible ways to select the leader.
9
From a group of 4 boys and 3 girls, find the number of ways to select 3 children with at least one boy included.
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Solution: Step 1: Calculate total ways to select 3 from 7: C(7,3) = 35 Step 2: Calculate ways to select 3 girls (no boys): C(3,3) = 1 Step 3: Subtract invalid case from total: 35 - 1 = 34
10
In a sports tournament, each team plays exactly one match against every other team. If 9 teams participate, how many matches are played in total?
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Solution: Step 1: Use combination formula C(n, 2) = n*(n-1)/2 Step 2: Plug in n = 9 teams Step 3: Calculate matches = 9*8/2 = 36 Step 4: Verify: Each team plays 8 matches, total pairs = 36
11
From a group of 8 individuals, determine the number of ways to select 2 members.
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Solution: Step 1: Use combination formula C(n, r) = n! / [r!(n - r)!] Step 2: Here, n = 8, r = 2 Step 3: Calculate C(8, 2) = 8! / [2!(8 - 2)!] Step 4: Simplify: 8! / (2! * 6!) = (8 * 7) / (2 * 1) Step 5: Compute: 56 ways
12
From a total of 8 individuals, a group of 4 is to be formed. (i) How many ways can the group be formed if 2 specific individuals must be included? (ii) How many ways can the group be formed if 2 specific individuals must be excluded?
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Solution: Step 1: Total individuals = 8, Group size = 4 Step 2: (i) 2 specific individuals included: Remaining to choose = 4 - 2 = 2 from 8 - 2 = 6 Step 3: Ways = 6 C 2 = (6 * 5) / (2 * 1) = 15 Step 4: (ii) 2 specific individuals excluded: Choose 4 from 8 - 2 = 6 Step 5: Ways = 6 C 4 = (6 * 5) / (2 * 1) = 15 Answer: 15 and 15
13
There are 8 pathways from City A to City B, and 6 pathways from City B to City C. How many distinct routes exist for traveling from City A to City C via City B?
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Solution: Step 1: Apply multiplicative rule for sequential routes Step 2: Calculate: 8 (A to B) Ɨ 6 (B to C) = 48 Step 3: Conclusion: Total distinct travel routes = 48
14
In a group of 30 individuals, if every person shakes hands with every other person exactly once, how many total handshakes occur?
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Solution: Step 1: Use the combination formula C(n, 2) = n(n-1)/2 where n = 30 Step 2: Calculate total handshakes: C(30, 2) = 30 * 29 / 2 Step 3: Simplify: 30 * 29 = 870 Step 4: Divide by 2: 870 / 2 = 435 Step 5: Total handshakes = 435
15
In how many ways can 4 students be selected from a group of 4 boys and 5 girls such that at least 2 boys are included?
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Solution: Step 1: Total students = 9 (4 boys + 5 girls) Step 2: Calculate total ways to select 4 students = ā¹Cā‚„ = 126 Step 3: Calculate invalid cases (0 boys or 1 boy) Step 4: Case 1: 0 boys (all girls) = āµCā‚„ = 5 Step 5: Case 2: 1 boy and 3 girls = ā“Cā‚ Ɨ āµCā‚ƒ = 4 Ɨ 10 = 40 Step 6: Total invalid cases = 5 + 40 = 45 Step 7: Valid cases = Total - Invalid = 126 - 45 = 81
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