Bank Math Master

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

Solution:

The word 'RUMOUR' consists of 6 letters in which each of 'R' and 'U' comes twice.

Here,
Number of letters, n = 6
Number of letter 'R', p = 2
Number of letter 'U', q = 2

∴ Number of arrangements
= n!/(p! × q!)
= 6!/(2! × 2!)
= (6 × 5 × 4  ×3 × 2 × 1)/(2 × 2)
= 180

Hence, The letters of the word 'RUMOUR' be arranged in 180 ways.

Question: In how many ways 7 pictures can be hung from 5 picture nails on a wall ?

Solution:

We have to find the number of ways to hang 7 pictures from 5 picture nails on a wall.
Now, we know that a permutation is the arrangement of a set of data in some specific order. If we have total number of 'n' datasets and we have to choose 'r' objects from the dataset.

Here,
n = 7
r = 5

Hence, the required number 
= ⁿP_r
= n!/(n - r)!
₌ 7!/(7 - 5)!
= 7!/2!
= (7 × 6 × 5 × 4 × 3 × 2 × 1)/(2 × 1)
= 2520

Hence, 7 pictures can be hung from 5 picture nails on a wall in 2520 ways.

Question: In how many ways can 5 balls can be chosen from 9 different balls?

Solution: 

Here,
Total number of different balls, n = 9
Chosen balls from different balls, r = 5

The number of ways 5 balls can be chosen is
= ⁿC_r
= n!/r!(n - r)!
= 9!/5!(9 - 5)!
= 9!/(5! × 4!)
= (9 × 8 × 7 × 6 × 5!)/(5! × 4!)
= (9 × 8 × 7 × 6)/4!
= 3024/(4 × 3 × 2 × 1)
= 3024/24
= 126

∴ 5 balls can be chosen from 9 different balls in 126 ways.

Question: Which of the following is not a leap year?

Solution:

To determine if a year is a leap year, you can use the following rules:

(1) If the year is evenly divisible by 4, go to step 2. If not, it is not a leap year.
(2) If the year is divisible by 100, go to step 3. If not, it is a leap year.
(3) If the year is divisible by 400, go to step 4. If not, it is not a leap year.
(4) If the year is divisible by 400, it is a leap year. Otherwise, it is not.

Now, let's apply these rules to the year 1700:

(1) 1700 is divisible by 4.
(2) 1700 is divisible by 100.
(3) 1700 is not divisible by 400.

According to the rules, since 1700 is divisible by 4 but not by 400, it is not a leap year.

Question: Today is Saturday. After 54 days, it will be:

Solution:

54 has to be divided by 7  [Because, 7 days = 1 week]



∴ The day after 54 days will be
= (Saturday + 5 days)
= Thursday

Question: How many times are the short hand and the long hand of a clock of a right angles during their motion from 1:00 pm to 10:00 pm?

Solution:

The duration from 1:00 pm to 10:00 pm is 9 hours.
During each of these 9 hours, the hands of the clock are at right angles twice.

So, required number = 9 × 2 = 18

Question: A committee is to consist of two members. If there are eight men and five women available to serve on the committee, how many different committees can be formed?

Solution:

Here,
Total members,
n = 8 + 5 = 13
r = 2

Number of committees can be formed
= ¹³C₂ 
= 13 × 12/2 × 1
= 156/2
= 78

∴ 78 different committees can be formed.

Question: If a 30 m ladder is placed against a 15 m wall such that it just reaches the top of the wall, then the elevation of the wall is equal to -

Solution:

Here,
Ladder, AC = 30m
Wall, AB = 15m
∠ACB = θ =?



sinθ = AB/AC  [ sinθ = লম্ব/অতিভুজ ]
⇒ sinθ = 15/30
⇒ sinθ = 1/2
⇒ sinθ = sin30^°  

∴ θ = 30^° 

Question: How many days are there in z weeks z days?

Solution:

1 week = 7 days
z weeks = 7z days

∴ z weeks z days = (7z + z) days = 8z days

Truth Table:

If z = 4

4 weeks + 4 days
= (7 × 4) + 4 [ Putting '4' in (7z + 4) ]
= 28 + 4
= 32

∴ 8z = 8 × 4 = 32

Question: A committee of 3 members is to be formed by selecting out of 5 men and 4 women. In how many different ways the committee can be formed if it should have 1 man and 2 women?

Solution:

Here,
1 man can be selected from 5 men in
= ⁵C₁
= 5/1
= 5 ways

2 women can be selected from 4 women in
= ⁴C₂
= (4 × 3)/(2 × 1)
= 12/2
= 6 ways

∴ The total number of ways the committee can be formed
= 5 × 6 ways
= 30 ways

Question: Between two book-ends in your study are displayed your five books. If you decide to arrange the five books in every possible combination and moved just one book every minute, how long would it take you?

Solution:

Here,
Number of ways of arranging 5 books
= 5!
= 5 × 4 × 3 × 2 × 1
= 120

So, total time taken
= 120 minutes [ 60 minutes = 1 hour ]
= 2 hours

Question: If there are 12 persons in a party and if each of them shake hands with each other, then number of hand shakes in party are:

Solution:

Two people can make 1 handshake.

∴ Number of handshakes
= ¹²C₂
= (12 × 11)/(2 × 1)
= 66

∴ The number of hand shakes in party are: 66

Question: The angle of elevation of a ladder leaning against a wall is 60° and the foot of the ladder is 4.6 m away from the wall. The length of the ladder is:

Solution:

Let AB be the wall and AC be the ladder.


Then,  ∠ACB = 60°  and BC = 4.6 m

BC/AC = cos60° [ cosθ = ভূমি/অতিভুজ ]
⇒ BC/AC = 1/2
⇒ AC = 2BC
⇒ AC = 2 × 4.6
∴ AC = 9.2m

∴ The length of the ladder is 9.2m

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

Solution:

The word 'LEADER' contains 6 letters.
It contains namely 2E.

Here,
Number of letters, n = 6
Number of letter 'E', p = 2

∴ The required number of ways
= n!/p!
= 6!/2!
= (6 × 5 × 4 × 3 × 2 × 1)/(2 × 1)
= 720/2
= 360

∴ The letters of the word 'LEADER' be arranged in 360 ways.

Question: Through what angle does the minute hand of a clock turn in 30 minutes?

Solution:

Angle traced by the minute hand in 30 minutes
= [(360/60) × 30]°
= 180°

The minute hand of a clock completes one full revolution in 60 minutes, as there are 60 minutes in an hour.
Therefore, in 30 minutes, the minute hand covers half of that full revolution.

So, the angle turned by the minute hand in 30 minutes is half of 360 degrees, which is 180 degrees.

Question: In how many ways 2 books can be chosen from the class of 20 books?

Solution:

Here,
n = 20
r = 2

The number of ways
= ⁿC_r 
= n!/r!(n - r)!
= 20!/2!(20 - 2)!
= 20!/2! × 18!
= 20 × 19 × 18!/2! × 18!
= 20 × 19/2 × 1
= 190

∴ 2 books can be chosen from the class of 20 books in 190 ways.

Question: ³C₂ - ³P₂ =?

Solution:

³C₂ - ³P₂ 
= (3 × 2/2 × 1) - {3!/(3 - 2)!}
= (6/2) - {3!/1!}
= 3 - (3 × 2)
= 3 - 6
= -3

∴ ³C₂ - ³P₂ = -3

Question: Find the number of triangles which can be formed by joining the angular points of a polygon of 8 sides as vertices.

Solution:

A triangle needs 3 points.
And a polygon of 8 sides has 8 angular points.

Hence,
Number of triangles formed
= ⁸C₃ 
= 8 × 7 × 6/3 × 2 × 1
= 336/6
= 56

∴ 56 triangles can be formed by joining the angular points of a polygon of 8 sides as vertices.

Question: In how many ways can a football team be chosen out of 14 players?

Solution:

A football team contains 11 players.

Required number of ways
= ⁿC_r 
= ⁿC_{n - r}
= ¹⁴C_14-11
= ¹⁴C₃
= (14 × 13 × 12)/(3 × 2 × 1)
= 364

∴ A football team be chosen out of 14 players in 364 ways.

Question: How many different six digit numbers can be formed using all of the following digits 2, 2, 5, 5, 5, 4?

Solution:
প্রদত্ত অঙ্ক মোট 6টি যার মধ্যে 2টি 2 এবং 3টি 5 আছে।

∴ নির্ণেয় ছয় অঙ্কবিশিষ্ট মোট গঠিত সংখ্যা = 6!/(2! × 3!) টি
= 60 টি 

Question: There are 12 true-false questions in an examination, these questions can be answered in-

Solution:

There are 12 true-false questions.

Each question can be answered in two ways i.e. true or false.

Therefore, the total number of ways is
= 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
= (2 × 2 × 2 × 2) × (2 × 2 × 2 × 2) × (2 × 2 × 2 × 2)
= 2⁴ × 2⁴ × 2⁴
= 16 × 16 × 16
= 4096

∴ There are 12 true-false questions in an examination, these questions can be answered in- 4096 ways.

Question: The angle between the long hand and the short hand of a clock when the time is 4.20, is:

Solution:

Here,
Long hand = Minute hand = 20 minutes
Short hand = Hour hand = 4 hours

Formula = ( 30 × Hours - 11/2 minutes)
= 30 × 4 - (11/2) × 20
= 120 - 110
= 10^° 

∴ The angle between the long hand and the short hand of a clock when the time is 4.20, is: 10°

Question: If 16^th March, 2005 is Wednesday, what was the day of the week on 16^th March, 2004?

Solution:
Since ¹⁶th March is coming after February leap year will not count in 2004. 
Odd day is 1.

∴ 16^th March, 2004 is Tuesday.