Bank Math Master

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

Solution:
Number of letter in word = 5
Repeated letter T = 2, and rest of the letters are unique.

∴ The number of arrangement = 5!/2! = 120/2 = 60

Question: It was Monday on Jan 1, 2018. What was the day of the week on Jan 1, 2022?

Solution: 
We know that,
The first day and the last day of a year are same day of the week, except leap year.
In a leap year, the week day of the last day of the year is next to the week day of the first day of the year.

Now,
It was Monday on Jan 1, 2018 so, it was Monday on Dec 31, 2018.
It was Tuesday on Jan 1, 2019 so, it was Tuesday on Dec 31, 2019.
It was Wednesday on Jan 1, 2020 (which was leap year) so, it was Thursday on Dec 31, 2020.
It was Friday on Jan 1, 2021 so, it was Friday on Dec 31, 2021.
It was Saturday on Jan 1, 2022.

Question: If 6^th March, 2005 is Sunday, what was the day of the week on 6^th March, 2004?

Solution:
Since 6^th March coming after February. Leap year will not count for 2004.
∴ There number of odd day is 1.

So it had to be Saturday, which was one day before Sunday 

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 ways
2 women can be selected from 4 women in ⁴C₂ = 6 ways

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

Question: A committee of 5 members is to be formed by selecting out of 7 men and 6 women. In how many different ways the committee can be formed if it should have at least 3 men?

Solution:
Men                    Women                  Ways
-------------------------------------------------------
3                          2                            ⁷C₃ × ⁶C₂ = 35 × 15 = 525
4                          1                            ⁷C₄ × ⁶C₁ = 35 × 6 = 210 
5                          0                            ⁷C₅ × ⁶C₀ = 21 × 1 = 21

∴ Total number of ways = 525 + 210 + 21 = 756

Question: An accurate clock shows 8 o'clock in the morning. Through how may degrees will the hour hand rotate when the clock shows 2 o'clock in the afternoon?

Solution:
Time between 8 o'clock to 2 o'clock = 6 hours

Hour hand rotate in 12 hours 360°
∴ Hour hand rotate in 6 hours (360° × 6)/12
= 180° 

Question: An observer 1.6 m tall is 20√3 m away from a tower. The angle of elevation from his eye to the top of the tower is 30°. The height of the tower is:

Solution:

Here, 
Height of the observer AB = 1.6 m 
Distance  between observer and tower AC = 20√3 m
∴ CE = AB = 1.6 m 
Height of the tower is CD = CE + DE.

From ΔBDE we get,
BE = AC = 20√3 m

Now,
tan30° = DE/BE 
⇒ 1/√3 = DE/(20√3)
⇒ DE = (20√3)/√3 
∴ DE = 20 

∴ The height of the tower is = CE + DE = 1.6 + 20 m = 21.6 m 

Question: The angle of elevation of the sun, when the length of the shadow of a tree √3 times the height of the tree, is:

Solution:

Let,
Height of the tree BC = x
Length of the shadow AB = √3x
The angle of elevation of the sun ∠C = θ

Now,
tanθ = BC/AB 
⇒ tanθ = x/(√3x)
⇒ tanθ = 1/√3
⇒ tanθ = tan30°
∴ θ = 30°

Question: A group of 7 members having a majority of boys is to be formed out of 7 boys and 4 girls. The number of ways can be formed is the group-

Solution:
Boys                   Girls                  Ways
-------------------------------------------------------
6                          1                            ⁷C₆ × ⁴C₁ = 7 × 4 = 28 
5                          2                            ⁷C₅ × ⁴C₂ = 21 × 6 = 126
4                          3                            ⁷C₄ × ⁴C₃ = 35 × 4 = 140


∴ Total number of ways = 28 + 126 + 140 = 294

Question: The angle between the minute hand and the hour hand of a clock when the time is 8 : 30 is:

Solution:
= │(11m - 60h)/2│°
= │(11 × 30 -  60 × 8)/2│°
= │(330 - 480)/2│°
= │- 150/2│°
= 75°

Question: At the end of a banquet 10 people shake hands with each other. How many handshakes will there be in total?

Solution:
Two people can make 1 handshake
∴ No. of handshakes = ¹⁰C₂ = 45

Question: How many unique ways are there to arrange the letters in the word PRETTY?

Solution:
Number of letter in word = 6
Repeated letter T = 2, and rest of the letters are unique.

∴ The number of arrangement = 6!/2! = 720/2 = 360

Question: A watch which gains 5 seconds in 3 minutes was set right at 7 a.m. In the afternoon of the same day, when the watch indicated quarter past 4 o'clock, the true time is:

Solution:
Time from 7 a.m. to 4 : 15 p.m. = 9 hrs 15 min. = 37/4 hrs.
Now,
3 min. 5 sec. of this clock = 3 min. of the correct clock.
⇒ 37/720 hrs. of this clock = 1/20 hrs. of the correct clock.
⇒ 37/4 hrs. of this clock = {(1/20) × (720/37) × (37/4)} hrs. of the correct clock.
= 9 hrs. 

∴ The correct time is 9 hrs. after 7 a.m. =  4 p.m.

Question: You need to put your reindeer, Ezekiel, Lancer, Rudy, and Jebediah, in a single-file line to pull your sleigh. However, Jebediah and Rudy are best friends, so you have to put them next to each other, or they won't fly. How many ways can you arrange your reindeer so that they can fly?

Solution:
We can count the number of arrangements where Jebediah and Rudy are together by treating them as one. 
So, All of the reindeer where Jebediah and Rudy are together can be arranged in 3! ways = 6 ways
And, Jebediah and Rudy can be arranged in 2! ways = 2 ways 

∴ Total number of ways of arrangements is = 6 × 2 = 12 ways 

Question: How many different words can we make using the letters A, B, E and L without repeating the letters?

Solution:
We have 4 choices for the first letter, 3 choices for the second letter, 2 choices for the third letter and 1 choice for the fourth letter. Hence the number of words is given by 4 × 3 × 2 × 1 = 4! = 24

Question: The calendar for the year 2007 will be the same for the year:

Solution:
Count the number of odd days from the year 2007 onwards to get the sum equal to 0 odd day.
Year : 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017 Odd day : 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1.
Sum of odd days = 14 odd days ≡  0 odd days.

 Calendar for the year 2018 will be the same as for the year 2007.

Question: In a certain country, the car number plate is formed by 4 digits from the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 followed by 3 letters from the alphabet. How many number plates can be formed if neither the digits nor the letters are repeated?

Solution:
Formed of number ⁹P₄ =3024
Formed of alphabet ²⁶P₃ = 15600

∴ Number of number plates = ⁹P₄  × ²⁶P₃ = 3024 × 15600 = 47174400