ব্যাংক নিয়োগ প্রস্তুতি ⎯ লং কোর্স

Question: What is the probability of getting 53 Tuesdays in a leap year?

Solution: 
1 year = 365 days . A leap year has 366 days
A year has 52 weeks. Hence there will be 52 Tuesdays for sure.
52 weeks = 52 x 7 = 364 days
366 – 364 = 2 days

In a leap year there will be 52 Tuesdays and 2 days will be left.

These 2 days can be:
1. Sunday, Monday
2. Monday, Tuesday
3. Tuesday, Wednesday
4. Wednesday, Thursday
5. Thursday, Friday
6. Friday, Saturday
7. Saturday, Sunday

Of these total 7 outcomes, the favourable outcomes are 2.

Hence the probability of getting 53 days = 2/7

প্রশ্ন: ⁵P₂ - ⁵C₂ =?

সমাধান: 
⁵P₂
= 5!/(5 - 2)!
= 5!/3!
= 20

⁵C₂
= 5!/2!(5 - 2)!
= 5!/2! 3!
= 10

∴ ⁵P₂ - ⁵C₂ = 20 - 10
= 10

Questoin: A sum was put at simple interest at a certain rate for 3 years. Had it been put at 1% higher rate, it would have fetched 510 taka more. The sum is - 

Solution: 
Let the sum be Tk x and original rate be R%
Then, ((x × (R +1) × 3)/100) - {(x × R × 3)/100} = 510
⇒ 3Rx + 3x - 3Rx = 51000
⇒ 3x = 51000
⇒ x = 17000

Hence, sum = Tk 17,000

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

Solution:
D is taken two times.

the word 'WEDDING' be arranged in = 7!/2!
= 5040/2
= 2520 

Question: Rakib and his wife appear in an interview for two vacancies in the same post. The probability of Rakib's selection is (1/7) and the probability of his wife's selection is (1/5). What is the probability that only one of them is selected?

Solution: 
the probability of Rakib's selection is (1/7)
the probability of Rakib's not selection is = 1 - (1/7)
= (7 - 1)/7
= 6/7

the probability of his wife's selection is (1/5)
the probability of his wife's not selection is = 1 - (1/5)
= (5 - 1)/5
= 4/5

the probability that only one of them is selected is = (1/7) × (4/5) + (1/5) (6/7)
= (4/35) + (6/35)
= (6 + 4)/35
= 10/35
= 2/7 

Question: A sum of money at compound interest doubles itself in 15 years. It will become eight times of itself in-

Solution: 
let the sum P 

2P = P (1 + r)¹⁵
⇒ (1 + r)¹⁵ = 2

let, sum will 4 times in n years
8P = P(1 + r)ⁿ
⇒ 8 = (1 + r)ⁿ
⇒ 2³ = (1 + r)ⁿ
⇒ ((1 + r)¹⁵)³ = (1 + r)ⁿ
⇒ (1 + r)⁴⁵ = (1 + r)ⁿ
∴ n = 45 years

Question: A committee of 3 members is selected out of 4 men and 3 women. What is the probabiity that the comittee has at least 1 man? 

Solution: 
৭ জন থেকে ৩ জন বাছাই করার উপায় = ⁷C₃
= 35 

৩ জন মহিলা থেকে ৩ জনই বাছাই করার উপায় = ³C₃
= 1

কমিটির ৩ জনই মহিলা হওয়ার সম্ভাবনা = 1/35

∴ কমপক্ষে ১ জন পুরুষ নিয়ে ৩ জনের কমিটি করার সম্ভাবনা = 1 - (1/35)
= (35 - 1)/35
= 34/35

Question: 6 people at a party shake hands once with everyone else in the room.How many handshakes took place?

Solution: 
handshakes took place = ⁶C₂
= 6!/2! 4!
= 15

Question: At what rate percent per annum will a sum of money double in 4 years?

Solution:
ধরি, সুদের হার r %

আসল x টাকা 
সুদাসল = 2x টাকা 
সুদ = 2x - x টাকা 
= x টাকা 

x = x × r × 4/100
∴ r = 25%

Question: What is the difference between the compound interests on Tk. 5000 for 1 year at 4% per annum compound yearly and half yearly?

Solution:
Interest compounded half yearly = 5000(1 + 2/100)² - 5000
= 5000(102/100)² - 5000
= 5000(1.02)² - 5000 .........(i)

As we know, A = P(1 + R/100)t
Interest compound yearly = 5000(1 + 4/100)¹ - 5000
= 5000(104/100) - 5000
= 5000(1.04) - 5000 ............(ii)

From (i) and (ii)
Required difference = 5000(1.02)² - 5000 - 5000(1.04)¹ + 5000
= 5000(1.0404) - 5000(1.04)
= 5202 - 5200
= 2

Question: A license plate begins with three letters. If the possible letters are A, B, C, D , how many different permutations of these letters can be made if no letter is used more than once?

Solution: 
For the first letter, there are 5 possible choices. After that letter is chosen, there are 4 possible choices. Finally, there are 3 possible choices.
4 × 3 × 2
= 24

Question: 500 shares, of par value TK. 20 each, are purchased from Rasel by Himel at a price of Tk. 25 each. Find the amount required to purchase the shares.

Solution: 
Face value of each share = Tk. 20
The market value of each share = Tk. 25
Number of shares = 500
Amount required to purchase the shares = 500 × 25
= 12500 tk.

Question: 1000 tk. is being charged at 50% per annum. what is the interest for 2nd year at compound interest?

solution:
১মবছরে চক্রবৃদ্ধি সুদাসল =  1000 (1 + 50/100)1 
= 1000 (1 + 1/2)¹
= 1000 (3/2)¹
= 1500 টাকা  

২য় বছরে চক্রবৃদ্ধি সুদাসল = 1000 (1 + 50/100)2
= 1000 (1 + 1/2)²
= 1000 (3/2)²
= 2250 টাকা 

২য় বছরের জন্য সুদ = 2250 - 1500 টাকা 
= 750 টাকা 
 

Question: How many different ways can 4 students line up to purchase a new laptop?

Solution: 
4 students line up to purchase a new laptop
= 4!
= 24

Question: Joni buys a 40 tk shares in a company, which pays 10% dividend. Joni buys the share at such a price that his profit is 16% on his investment. At what price did Joni buy the share?

Solution: 
Dividend (profit) given by the company on 1 share = 10% of  40 = 4 tk.

Suppose the man buys one share for x.

Therefore, Joni’s profit = 16% of  x = 16x/100
According to the problem, 16x/100 = 4
⟹ x = 25

Joni bought the share at  25 tk.

Question: Mamun buys 1,000 shares of stock for the current stock price of 20 tk. per share.  If the stock price goes up to 25 tk. per share, by what percentage does Mamun increase his money?

Solution: 
প্রতি শেয়ার ২০ টাকা হলে, মোট শেয়ার = ১০০০ × ২০ টাকা 
= ২০০০০ টাকা 

২৫ টাকা হলে মোট শেয়ার = ১০০০ × ২৫ টাকা 
= ২৫০০০ টাকা 

টাকা বৃদ্ধি পাবে = ২৫০০০ - ২০০০০ টাকা 
= ৫০০০ টাকা 

শতকরা বৃদ্ধি = (৫০০০/২০০০০) × ১০০% 
= ২৫% 

Question: How many 4-digit numbers can be formed from the digits 3, 5, 6, 7 and 9, which are divisible by 5 and none of the digits is repeated?

Solution:
Number will be divisible by 5 if the last number is 5.
So, first number can be chosen in ⁴C₁ ways 
= 4 ways
As the digit is not repeated
second number can be chosen in ³C₁ 
= 3 ways

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

∴ Total ways = 4 × 3 × 2 ways
= 24 ways

Question: An amount of Tk 10000 becomes Tk 11025 in 1 year if the interest is compounded half-yearly. What is the rate of compound interest per annum?

Solution:
Let the compound interest be r%

ATQ,
10000 × {1 + (r/2)}² = 11025
⇒ {(2 + r)/2}² = 11025/10000
⇒ (2 + r)/2 = 105/100 [applying square root]
⇒ 2 + r  = 210/100
⇒ 200 + 100r = 210
⇒ 100r = 10
⇒ r = 10/100
⇒ r = (10/100%) × 100
⇒ r  = 10%

Question: A sum of Tk. 10,000 amounts to Tk. 12,000 in 4 years at the rate of simple interest. What is the rate of interest?

Solution: 
সুদ = ১২০০০ - ১০০০০ টাকা 
= ২০০০ টাকা 

ধরি, সুদের হার r%  

I = pnr
⇒ 2000 = 10000 × 4 × r/100
⇒ r = 5

সুদের হার ৫%

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

Solution: 
Number of fingers n = 4
Number of rings r = 6
∴ 6 rings may be worn in = 4⁶  ways.

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

Solution:
the given words contain 5 diffrerent letters.

∴ they can be arranged in = 5! ways
= 120 ways 

Question: If a certain coin is flipped, the probability that the coin will land heads is 1/2. If the coin is flipped 3 times, what is the probability that it will land heads up on the first 2 flips and not on the last flip?

Solution: 
The probability of landing heads and not landing on heads is same = 1/2
The probability of first two heads =(1/2) × (1/2)
The probability of last  landing not on heads = 1/2
The total probability =(1/2) × (1/2) × (1/2) 
= 1/ 2³
= 1/8

Question: A question paper has two parts, A and B, each containing 5 questions. If a student has to choose 3 from part A and 2 from part B, in how many ways can he choose the questions?

Solution:
ways to choose 3 from part A = ⁵C₃
ways to choose 2 from part B = ⁵C₂

choose 3 from part A and 2 from part B = ⁵C₃ × ⁵C₂
= {5!/(3! 2!)} × {5!/(2! 3!)}
= 10 × 10
= 100

Question: A box contains 5 white balls, 3 black balls and 4 red balls. In how many ways can 3 balls be drawn from the box, if at least one white ball is to be included in the draw?

Solution:
Total number of balls = 5 + 3 + 4 
= 12 balls

choosing 3 balls from 12 balls = ¹²C₃
= 220 ways

choosing three balls from black and red balls = ⁷C₃
= 35 ways 

∴ ways can 3 balls be drawn from the box, if at least one white ball is to be included in the draw is = 220 - 35 ways
= 185 ways

Question: In a lottery, there are 10 prizes and 15 blanks. A lottery is drawn at random. What is the probability of getting a prize?

Solution:
Total outcome = (10 + 15)
= 25
favorable outcome = 10

∴ the probability of not getting a prize is = 10/25
= 2/5