Unit5 - Subjective Questions
PEA305 • Practice Questions with Detailed Answers
State and explain the Fundamental Principle of Multiplication and the Fundamental Principle of Addition with one example for each.
1. Fundamental Principle of Multiplication:
If an event can occur in different ways, and following it, another event can occur in different ways, then the total number of ways in which both events can occur in that order is .
- Example: If a person has 3 shirts and 2 pairs of pants, the total number of different outfits they can wear is .
2. Fundamental Principle of Addition:
If there are two jobs such that they can be performed independently in and ways respectively, then either of the two jobs can be performed in ways.
- Example: If a student can choose a project from 3 science topics or 4 history topics, the total number of ways to choose one topic is .
In how many ways can the letters of the word 'MATHEMATICS' be arranged? Explain the treatment of repeated letters.
To find the number of arrangements, we count the total letters and the frequency of each repeating letter.
Step 1: Count total letters
The word 'MATHEMATICS' has 11 letters.
Step 2: Identify repeated letters
- M appears 2 times.
- A appears 2 times.
- T appears 2 times.
- H, E, I, C, S appear 1 time each.
Step 3: Apply the formula for permutation with repetition
The formula is , where is the total number of items and represent the count of identical items.
Answer: There are 4,989,600 distinct ways to arrange the letters.
A committee of 5 persons is to be selected from 6 men and 4 women. In how many ways can this be done if:
- There is no restriction?
- At least one woman is included?
Total Men = 6, Total Women = 4. Total People = 10.
We need to select 5 people.
Case 1: No restriction
We simply select 5 out of 10.
Case 2: At least one woman included
It is easier to calculate the total ways and subtract the case where no women are included (i.e., all 5 are men).
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Total ways: 252
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Ways to select 5 men only (from 6 men):
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Ways with at least one woman:
Answer:
- 252 ways.
- 246 ways.
Find the rank of the word 'MOTHER' if all the letters of the word are arranged in dictionary order.
To find the rank of 'MOTHER', we arrange the letters alphabetically: E, H, M, O, R, T.
Total letters = 6.
Step 1: Words starting with E
Fix E at the start. Remaining 5 letters can be arranged in ways.
Step 2: Words starting with H
Fix H at the start. Remaining 5 letters can be arranged in ways.
Step 3: Words starting with M
Our word starts with M. We fix M.
Next alphabetical letter is E. Words starting with ME: ways.
Next is H. Words starting with MH: ways.
Next is O. Our word matches MO. Fix O.
- Current Prefix: MO.
- Remaining letters: E, H, R, T.
Step 4: Continue inside MO...
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Next alphabetical is E. Words starting MOE: ways.
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Next is H. Words starting MOH: ways.
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Next is R. Words starting MOR: ways.
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Next is T. Matches MOT. Fix T.
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Current Prefix: MOT.
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Remaining letters: E, H, R.
Step 5: Continue inside MOT...
- Next is E. Matches MOTE. Fix E.
- Remaining: H, R.
- Alphabetical order is H then R. This forms MOTEHR (1 way).
- Next is MOTHER (The target word).
Calculation:
Rank =
Rank = .
Answer: The rank is 308.
Differentiate between Permutation and Combination with appropriate examples.
1. Definition:
- Permutation: Refers to the arrangement of objects in a specific order. The sequence matters.
- Combination: Refers to the selection of objects where order does not matter.
2. Formulas:
- Permutation:
- Combination:
3. Key Difference:
- In permutation, AB is different from BA.
- In combination, AB is the same as BA.
4. Examples:
- Permutation: Finding the number of ways to seat 3 students (A, B, C) in a row. (ABC, ACB, BAC, BCA, CAB, CBA are all distinct).
- Combination: Selecting a team of 3 students from a group of 10. Selecting A, B, and C is the same team as selecting C, B, and A.
How many numbers greater than 2000 can be formed using the digits 1, 2, 3, 4, 5 without repetition?
The available digits are {1, 2, 3, 4, 5}. Total digits = 5.
Since we need numbers greater than 2000, we can have 4-digit numbers or 5-digit numbers.
Case 1: 5-digit numbers
Any 5-digit number formed by these digits is greater than 2000.
Total permutations = .
Case 2: 4-digit numbers
For a 4-digit number to be greater than 2000, the first digit (thousands place) must be 2, 3, 4, or 5.
- Thousands place: 4 options (2, 3, 4, 5).
- Remaining 3 places: We have 4 digits left, to be arranged in 3 spots.
- Calculation: .
Total Numbers:
Answer: 216 numbers.
Explain the concept of Circular Arrangement. In how many ways can 6 people be seated around a round table? How does this change if we are arranging beads in a necklace?
Concept of Circular Arrangement:
In a linear arrangement, the order is defined by a start and an end. In a circle, there is no fixed starting point. Therefore, relative positions matter. The number of ways to arrange distinct objects in a circle is .
1. Seating 6 people at a round table:
Since relative order matters and clockwise/anti-clockwise arrangements are distinct for people:
2. Arranging beads in a necklace:
For a necklace or garland, the arrangement can be flipped over. Therefore, the clockwise and anti-clockwise arrangements are indistinguishable (the same). The formula is .
Calculate the number of diagonals in a Decagon (a polygon with 10 sides) using the principles of combination.
A diagonal is formed by connecting any two non-adjacent vertices of a polygon.
Step 1: Total lines connecting vertices
A decagon has vertices. The total number of ways to select 2 vertices to draw a line is .
Step 2: Subtract the sides
The lines connecting vertices include both the diagonals and the sides of the polygon. We must subtract the sides.
Number of sides .
Step 3: Calculate Diagonals
Answer: A decagon has 35 diagonals.
Define Mutually Exclusive Events and Independent Events in probability with symbolic representation.
1. Mutually Exclusive Events:
Two events and are said to be mutually exclusive (or disjoint) if they cannot occur at the same time. If event happens, event cannot, and vice versa.
- Symbolically: .
- Addition Rule: .
2. Independent Events:
Two events and are independent if the occurrence (or non-occurrence) of one event does not affect the probability of the occurrence of the other.
- Symbolically: and .
- Multiplication Rule: .
Three unbiased coins are tossed simultaneously. Find the probability of getting:
- Exactly two heads
- At least two heads
- No heads
Sample Space (S):
Total outcomes = .
1. Exactly two heads:
Event . Favorable outcomes = 3.
2. At least two heads:
This means getting 2 heads OR 3 heads.
Event . Favorable outcomes = 4.
3. No heads:
This means getting all tails.
Event . Favorable outcomes = 1.
Two dice are rolled simultaneously. Find the probability that the sum of the numbers on the dice is:
- Equal to 8
- A multiple of 4
Total outcomes: When two dice are rolled, total outcomes .
1. Sum equal to 8:
Let be the event where the sum is 8.
Pairs: .
Number of favorable outcomes = 5.
2. Sum is a multiple of 4:
Possible sums are 4, 8, 12 (since max sum is 12).
- Sum = 4: 3 pairs.
- Sum = 8: 5 pairs.
- Sum = 12: 1 pair.
Total favorable outcomes = .
From a well-shuffled pack of 52 cards, two cards are drawn at random. Find the probability that both are Kings.
Total outcomes:
The number of ways to choose 2 cards from 52 is:
Favorable outcomes:
There are 4 Kings in a deck. We need to choose 2 of them.
Probability:
Simplify the fraction:
Answer: The probability is .
Explain the concept of Conditional Probability. State the formula for and solve the following: If , , and , find .
Concept:
Conditional probability is the probability of an event occurring given that another event has already occurred. reads as "Probability of A given B".
Formula:
Similarly,
Problem Solution:
Given:
We need to find :
Note that .
Answer: .
A bag contains 5 Red, 4 Green, and 3 Blue balls. Three balls are drawn at random. What is the probability that they are of different colors?
Total Balls: balls.
Step 1: Total possible outcomes
Selection of 3 balls from 12:
Step 2: Favorable outcomes
For the balls to be of different colors, we must select 1 Red, 1 Green, and 1 Blue.
Step 3: Calculate Probability
Answer: The probability is .
A problem in mathematics is given to three students A, B, and C whose chances of solving it are , , and respectively. What is the probability that the problem will be solved?
Let , , and be the probabilities that students A, B, and C solve the problem.
.
The problem is solved if at least one of them solves it. It is easier to calculate the probability that none of them solve it and subtract from 1.
Step 1: Probability of NOT solving
Step 2: Probability that NONE solve it
Since the events are independent:
Step 3: Probability problem is solved
Answer: The probability is .
What is the probability of getting 53 Sundays in a Leap Year?
Analysis of a Leap Year:
A leap year has 366 days.
52 Weeks:
52 weeks guarantee 52 Sundays. The existence of the 53rd Sunday depends on the remaining 2 extra days.
Possible combinations for the 2 extra days:
The consecutive days can be:
- Monday, Tuesday
- Tuesday, Wednesday
- Wednesday, Thursday
- Thursday, Friday
- Friday, Saturday
- Saturday, Sunday
- Sunday, Monday
Total possible outcomes = 7.
Favorable outcomes:
We need a Sunday. The favorable cases are:
- (Saturday, Sunday)
- (Sunday, Monday)
Total favorable outcomes = 2.
Probability:
In a group of 15 boys, there are 6 scouts. In how many ways can 10 boys be selected so as to include at least 4 scouts?
Total Boys = 15. Scouts = 6. Non-Scouts = 9.
Selection size = 10 boys.
Condition: At least 4 scouts.
Possible scenarios:
- 4 Scouts and 6 Non-Scouts
- 5 Scouts and 5 Non-Scouts
- 6 Scouts and 4 Non-Scouts
Calculation:
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Case 1:
Ways = -
Case 2:
Ways = -
Case 3:
Ways =
Total Ways:
Answer: 2142 ways.
Define Odds in Favor and Odds Against an event. If the odds against an event are 5:3, what is the probability of the occurrence of the event?
Definitions:
- Odds in Favor: If an event has favorable outcomes and unfavorable outcomes, odds in favor are or .
- Odds Against: The ratio of unfavorable outcomes to favorable outcomes, i.e., or .
Relationship with Probability:
If Odds against = , then:
Problem:
Odds against = 5:3. Here (unfavorable), (favorable).
Total outcomes = .
Probability of occurrence:
Find the number of arrangements of the letters of the word 'INDEPENDENCE'. In how many of these arrangements do all the vowels occur together?
Total letters in INDEPENDENCE: 12.
Letter counts: N=3, E=4, D=2, I=1, P=1, C=1.
1. Total Arrangements:
2. Vowels Together:
Vowels are: I, E, E, E, E (Total 5 vowels).
Consonants are: N, D, P, N, D, N, C (Total 7 consonants).
Treat the group of 5 vowels as 1 unit.
Total entities = 7 (consonants) + 1 (vowel group) = 8 units.
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Arranging the 8 units:
Among these 8 units, the consonants N (3 times) and D (2 times) repeat.
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Arranging the vowels inside the group:
The vowels are I, E, E, E, E. E repeats 4 times.
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Total Ways:
Answer: 16,800 arrangements.
There are 12 points in a plane, out of which 5 are collinear. Find the number of triangles that can be formed by joining these points.
Principle: A triangle is formed by selecting 3 non-collinear points.
Step 1: Total combinations
If no points were collinear, we could select any 3 points from 12.
Step 2: Subtraction for collinear points
5 points are collinear. If we select 3 points from these 5, they form a line, not a triangle. We must subtract these cases.
Step 3: Calculate valid triangles
Answer: 210 triangles can be formed.