ব্যাংক নিয়োগ প্রস্তুতি ⎯ লং কোর্স

Question: In how many different way can the letters of the word "ORANGE" be arranged?

Solution:
the given words contain 6 diffrerent letters.

∴ they can be arranged in = 6! ways
= 720 ways

Question: Three unbiased coins are tossed. What is the probability of getting at least 1 heads?

Soluttion:
Here total cases = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Total number of events = 8

The events of getting at least two heads = {HHH, HHT, HTH, HTT, THH, THT, TTH}
Number of expected events = 7

∴ Required probability = 7/8 

Question: In how many ways can be a group of 5 men and 2 women be made out of a total of 9 men and 3 women?

Solution:
There are 9 men and 3 women.
We have to select 5 men out of 9 and 2 women out of 3.

∴ The number of ways of making the selection = ⁹C₅ × ³C₂
= 126 × 3
= 378 ways.

Question: In a simultaneous throw of two dice, what is the probability of getting a total of 7?

Solution:
If two dices are thrown total events = 6² = 36
Event of getting a total of 7 = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}
Expected events = 6 

∴ Probability = 6/36 = 1/6

Question: In how many different ways can the letters of the word "BINARY" be arranged so that the vowels always come together?

Solution:
the given words contain 6 different letters.
When the vowels "ia" are taken together, we may treat them as 1 letter.

5 numbers can be arranged in = 5! ways
= 120 ways

two vowels can be arranged = 2! ways
= 2 ways

∴ Total number of arrangement = (120 × 2) ways
= 240 ways

Question: Once card drawn from a pack of 52 cards. What is the probability that the card drawn is either a red card or a king?

Solution:
Total card = 52
Total red card 26 and total king card = 4
but 26 red cards have also 2 king card,  so other 2 black king card.

So, Red card or king = 26 + 2 = 28

∴ Probability = 28/52 = 7/13

Question: In a party every person shakes hands with every other person. If there are 78 hands shakes, find the number of person in the party.

Solution:
Let n be the number of persons in the party
Number of hands shake = 78
Total number of hands shake is given by = ⁿC₂
Now,
According to the question,
ⁿC₂ = 78
⇒ n!/{2!(n - 2)!} = 78
⇒ {n × (n - 1)}/2 = 78
⇒ n^{2 _-} n =156
⇒ n² - 13n + 12n - 156 = 0
⇒ n(n - 13) + 12(n - 13) = 0
or, n = 13, -12

But, we cannot take negative value of n
So, n = 13
∴ number of persons in the party = 13

Question: In a box, there are 7 red, 8 blue and 9 green balls. One ball is picked up randomly. What is the probability that it is neither red nor green?

Solution:
Total number of balls, n(S) = (8 + 7 + 6) = 24

Let,
E = event that the ball drawn in neither red nor green = even that the ball drawn in blue.
∴ n(E) = 8

∴ P(E) = n(E)​/n(S)
= 8/24
​= 1/3

Question: How many 4-digit numbers can be formed from the digits 2, 4, 5, 7, and 8, which are divisible by 4, and none of the digits is repeated?

Solution:
A number is divisible by 4 if the last two digits of the number form a number divisible by 4.
The valid two-digit numbers divisible by 4, using the digits 2, 4, 5, 7, and 8 are: 24, 28, 48, 52, 72, 84 = 6 ways

So, first number can be chosen in = ³C₁ ways
= 3 ways

As the digit is not repeated second number can be chosen in = ²C₁
= 2 ways

∴ Total ways = (6 × 3 × 2) ways
= 36 ways

Question: In a lottery, there are 20 prizes and 25 blanks. A lottery is drawn at random. What is the probability of getting a prize?

Solution:
In a lottery, there are 20 prizes and 25 blanks,
that means (20 + 25) or 45 positions exist 20 prizes and 25 blanks.

∴ The probability of getting a prize,
= 20/45
= 4/9

Question: If ⁶P_r = 360 and If ⁶C_r = 15, then r =?

Solution:
ⁿP_r = ⁿC_r × r!
⇒ ⁶P_r = 15 × r!
⇒ 360 = 15 × r!
⇒ r! = 360/15
⇒ r! = 24
⇒ r! = 4 × 3 × 2 × 1
⇒ r! = 4!
∴ r = 4

Question: According to meteorological records, it rained on 14 days in the month of september last year. What is the probability that it will rain on fourth of september this year?

Solution:
September month has 30 days
favorable events = 14 days

∴ the probability that it will rain on fourth of september this year = 14/30
= 7/15

Question: If ¹⁶P_{r - 1} : ¹⁵P_{r - 1} = 16 : 7 then find r

Solution:

Question: If x and y are two positive integers and x + y = 5 then, what is the probability of x equals to 1?

Solution:
total possible ways = (1, 4), (2, 3), (3, 2), (4, 1) = 4
favorable event = (1, 4) = 1

∴ probability = 1/4

Quiestion: In how many ways can a cricket eleven be chosen out of 15 players?

Solution:
Required number of ways = ¹⁵C₁₁
= ¹⁵C_{(15 - 11)}
= ¹⁵C₄
= (14 × 13 × 12 ×11) /(4 × 3 × 2)
= 1365

Question: In how many ways can five different rings be worn on three fingers of one hand?

Solution:
Number of fingers  = 3
Number of rings = 5

∴ 5 rings may be worn in = 3⁵ ways.
= 243 ways

Question: In how many ways the letters of the word 'DIFFERENT' can be arranged?

Solution:
Total nnumber of letters in the word 'DIFFERENT' = 9
Repeating letters:
F = 2 times
E = 2 times
∴ Required no. of ways = 9!/(2! × 2!)
= 90720

Question: A select group of 4 is to be formed from 8 men and 6 women in such a way that the group must have at least 1 women. In how many different ways can it be done?

Solution:
The required number of ways =  ⁸C₃ × ⁶C₁ + ⁸C₂ × ⁶C₂ + ⁸C₁ × ⁶C₃ + ⁶C₄
= 56 × 6 + 28 × 15 + 8 × 20 + 15
= 336 + 420 + 160 + 15
= 931

Question: In how many ways can 6 examination papers be arranged so that the best and the worst papers never come together?

Solution:
Total ways = 6!
= 720 ways

if two papers come together, we can consider them one.
ways that they will come together = 5! × 2!
= 120 × 2
= 240 ways

∴ ways the best and the worst papers never come together = 720 - 240 ways
= 480 ways

প্রশ্ন: ⁶P₂ - ⁶C₂ =?

সমাধান:
⁶P₂
= 6!/(6 - 2)!
= 6!/4!
= 30

⁶C₂
= 6!/2!(6 - 2)!
= 6!/2! 4!
= 15

∴ ⁶P₂ - ⁶C₂ = 30 - 15
= 15

Question: The ratio of red balls, to yellow balls, to green balls, to blue balls in a basket is 2 : 3 : 4 : 5. What is the probability that a ball chosen at random from the basket is a red ball?

Solution:
The ratio of red balls, to yellow balls, to green balls in a basket is = 2 : 3 : 4 : 5
let, there are 2x red balls, 3x yellow balls, 4x green balls and 5x blue balls.

∴ Total balls = 2x + 3x + 4x + 5x
= 14x

∴ probability that a ball chosen at random from the basket is a red ball = 2x/14x
= 1/7

Question: In how many different ways can the letters of the word "ACTIVE" be arranged so that the vowels occupy only odd positions?

Solution:
The word "ACTIVE" consists of the letters, Vowels are (A, I, E) and Consonants (C, T, V)

The number of ways to arrange the 3 vowels in the 3 odd positions is = 3!
= 6

The number of ways to arrange the 3 consonants in the 3 even positions is = 3! = 6

∴ The requried ways = (6 × 6)
= 36

Question: Find the number of triangles which can be formed by joining the angular points of a polygon of 7 sides as vertices.

Solution:
the number of triangles which can be formed by joining the angular points of a polygon of 7 sides as vertices
= ⁷C₃
= 7!/(3! × 2!)
= 35 ways

Question: Five persons, A, M, J, R, and P, sit randomly in five chairs in a row. What is the probability that R and M sit next to each other?

Solution:
Total possibilities = 5! = 120
favorabole events = (4! × 2!)
= (24 × 2)
= 48

∴ probability = 48/120
= 2/5

Question: In how many different ways can the letters of the word "DESIGN" be arranged so that the vowels are at the two ends?

Solution:
The given word "DESIGN" contains 4 consonants and 2 vowel. 
At the two ends the two vowels can be arranged in 2! = 2 ways.
Remaining 4 letters can be arranged in = 4!
= 24 

∴ Total number of ways = (24 × 2) = 48