ব্যাংক নিয়োগ প্রস্তুতি ⎯ লং কোর্স

Question: A fair die is thrown once. What is the probability of getting a prime number?    
Solution: 
A standard fair die has 6 faces numbered: 1, 2, 3, 4, 5, 6
The prime numbers from 1 to 6 are: 2, 3, 5

Probability = (Number of favorable outcomes) / (Total number of outcomes)
= 3/6
= 1/2

Question: How many ways can 10 letters be posted in 5 post boxes, if each of the post boxes can take more than 10 letters?

Solution: 
Since all letters are independent, 
Total number of ways = 5¹⁰

Question: A bag contains 3 red balls and 2 blue balls. If two balls are drawn without replacement, what is the probability that both are red?

Solution: 
Total balls = 3 red + 2 blue = 5 balls. 
Probability that the first ball is red = 3/5

∴ Total probability (both red) = 3/5 × 1/2
= 3/10 

Question: Find the total number of distinct vehicle numbers that can be formed using two letters followed by two numbers. Letters need to be distinct.

Solution: 
Total number combinations = 10 × 10 = 100 (since digits can repeat)

Total number of vehicle numbers = 26 × 25 × 100 = 65000

Question: In how many ways can 5 people be seated around a circular table?

Solution: 
For n people around a circle, the number of distinct arrangements is = (n -1)!
So, for 5 people = (5 − 1)! = 4! = 24

Question: A committee of 3 men and 2 women is to be formed from 6 men and 4 women. How many ways can this be done?

Solution:
Ways to choose 3 men out of 6 = ⁶C₃ = 20
Ways to choose 2 women out of 4 = ⁴C₂ = 6

Total number of ways = 20 × 6 = 120

Question: A card is drawn from a deck of 52 cards, then replaced, and another card is drawn. What is the probability that both are aces?

Solution: 
Since the card is replaced, the events are independent. 

P(Ace on 1st draw) × P(Ace on 2nd draw) = (4/52) × (4/52) = (1/13)² = 1/169

Question: In how many ways can 12 identical balls be distributed among 4 distinct boxes such that each box contains at least one ball?

Solution: 

Now, distributing 8 identical balls among 4 boxes with no restrictions. 
Formula = ^{n + k - 1}C_{k - 1} ;where n = 8, k = 4
= ¹¹C₃ 
= 11!/(3! × 8!)
= (11 × 10 × 9)/(3 × 2 × 1)
= 990/6
= 165

Question: A box contains 10 red marbles and 15 blue marbles. If 5 marbles are drawn without replacement, what is the probability of getting exactly 2 red marbles?

Solution: 

Number of ways to choose 2 red marbles = ¹⁰C₂ = 45
Number of ways to choose 3 blue marbles = ¹⁵C₃ = 455

Favorable outcomes = 45 × 455 = 20475
Total possible outcomes = ²⁵C₅ = 53130

Probability = 20475/53130 = 0.3853 = 0.39

Question: Four boys and four girls are to be seated alternately around a round table. In how many different ways can this be done?

Solution: 

Total arrangements = 6 × 24 = 144

Question: Two dice are rolled. What is the probability that the sum is 7?

Solution: 
When two dice are rolled, the total number of possible outcomes is = 6 × 6 = 36
Favorable Outcomes (Sum = 7) = 6 [(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)] 

Probability = 6/36 = 1/6 

Question: How many ways can the letters in "TRIANGLE" be arranged if vowels must occupy odd positions?

Solution: 

Number of ways to place 3 vowels in 4 odd positions = ⁴C₃ = 4
Arrange 3 vowels in chosen positions = 3! = 6
Arrange 5 consonants in the remaining 5 positions = 5! = 120

Total arrangements = 4 × 6 × 120 = 2880

Question: A committee of 4 people is to be formed from 6 men and 4 women. How many ways can this be done if exactly 2 must be women?

Solution: 
2 women from 4 = ⁴C₂ = 6 
2 men from 6 = ⁶C₂ = 15 

Total number of ways = 6 × 15 = 90

Question: How many 4-letter words can be formed using the letters A, B, C with repetition allowed?

Solution: 

Question: A team of 5 players is to be selected from 7 forwards and 4 defenders. How many ways can this be done if exactly 3 forwards must be selected?

Solution: 
Ways to choose 3 forwards from 7 = ⁷C₃ = 35
Ways to choose 2 defenders from 4 = ⁴C₂ = 6

Total combinations = 35 × 6 = 210

Question: A fair coin is tossed 5 times. What is the probability of getting exactly 3 heads?

Solution: 
The number of ways to choose 3 Heads out of 5 tosses is = ⁵C₃ = 10 

∴ Probability of getting exactly 3 heads = 10/32 = 5/16

Question: How many distinct permutations can be formed with the letters of “BALLOON”?

Solution: 

∴ Total permutations = 7!/(2! × 2!) = 1260

Question: A password consists of 2 letters (A-Z only) followed by 2 digits (0-9). How many such passwords can be formed if no repetition is allowed?

Solution: 
Total passwords = 650 × 90 = 58500

Question: In how many ways can 5 different colored flags be arranged on a pole?

Solution: 
Total ways = 5!
= 120 

Question: A 4-digit number is formed using the digits 1 to 7 without repetition. What is the probability that the number is even?

Solution: 
Total available digits = 7 (1, 2, 3, 4, 5, 6, 7)
So, total 4-digit numbers = ⁷P₄ = 840 

The last digit must be even (2, 4, 6) = 3 choices.
Choosing 3 digits from 6 digits = ⁶P₃ = 120 

So, Number of even 4-digit numbers = 3 × 120 = 360

∴ Probability = 360/840 = 3/7 

Question: In how many ways can 3 boys and 3 girls be arranged in a line if all boys must stand together?

Solution: 

Since all 3 boys must stand together, we can consider them as one combined unit.

Number of ways to arrange these 4 units (3 girls + 1 group of boys) = 4! = 24
Number of arrangements among the 3 boys = 3! = 6

∴ Total arrangements = 24 × 6 = 144

Question: An urn contains 2 red, 3 green, and 2 blue balls. If 2 balls are drawn at random, find the probability that neither ball is blue.

Solution: 
Total number of balls = (2 + 3 + 2)
= 7

Let, E be the event of drawing 2 non-blue balls.

Then,
n (E) = ⁵C₂
= (5 × 4)/(2×1)
= 10

And, n (S) = ⁷C₂
= (7 × 6)/(2 × 1)
= 21

∴ P(E) = n(E)/n(S) = 10/21

Question: What is the probability that a randomly chosen 4-digit number has all distinct digits?

Solution: 
Total 4-digit numbers = 9999 − 1000 + 1 = 9000

A 4-digit number has digits in the form: ABCD, where:

A (thousands place): Can be 1-9 (9 options, cannot be 0).

B (hundreds place): Can be 0-9, except A (9 options).

C (tens place): Can be 0-9, except A and B (8 options).

D (units place): Can be 0-9, except A, B, and C (7 options).

 

Total numbers with distinct digits = 9 × 9 × 8 × 7 = 4536

∴ Probability = 4536/9000 = 63/125 

Question: The probability that a card drawn from a pack of 52 cards will be a diamond or a king is -

Solution: 
Here, n(S) = 52
There are 13 cards of diamonds (including one king) and there are three more kings.
Let E = the event of getting a diamond or a king
Then, n(E) = (13 + 3) = 16

∴ P(E) = n(E)/n(S)
= 16/52
= 4/13

Question: From a group of 10 people, how many ways can a president, vice-president, and secretary be chosen?

Solution: 
For selecting 3 different positions from 10 people,
Number of ways = ¹⁰P₃ ​
= 10 × 9 × 8 = 720