Q‒1. [2×10 marks] In the questions below suppose that a “word” is any string of seven letters of the alphabet, with repeated letters allowed. a) How many words begin with R and end with T?
b)How many words begin with A or B?
c) How many words begin with A or end with B?
d)How many words begin with A or B and end with A or B?
e) How many words begin with A or B or end with A or B?
f)How many words begin with a vowel and end with a vowel?
g) How many words begin with a vowel or end with a vowel?
h)How many words begin with AAB in some order?
i)How many words have no vowels?
j)How many words have exactly one vowel?

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Explanation:

Number of alphabets = 26

Number of vowels = 5

Number of strings in word = 7

A)How many words begin with R and end with T

First and last has been filled, positions left to fill : 7 - 2 = 5

26^5 = 11881376

B) How many words begin with A or B

First letter can begin with either A or B

Position left to fill = 7 - 1 = 6

2 options for first letter

2 × (26^6) = 617831552

C.) How many words begin with A or end with B

26^6 + 26^6 - 26^5 = 605950176

D.) How many words begin with A or B and end with A or B

Last and first positions have been filled either with A or B

Positions left to fill = 7-2 = 5

4 × (26^5) = 47525504

E) How many words begin with A or B or end with A or B?

(2 × 26^6 + 2 × 26^6 - 4 × 26^5) = 1188137600

f) How many words begin with a vowel and end with a vowel?

2^5 = 32

g) How many words begin with a vowel or end with a vowel?

(5×26^6) + (5×26^6) - (10×26^5) = 2970344000

h) How many words begin with AAB in some order?

AAB = 3 letters

Positions left to fill = 7 - 3 = 4

3 × 26^4 = 1,370,928

i) How many words have no vowels?

26 - 5 = 21 ; 21^7 = 1801088541

j) How many words have exactly one vowel

Number of words × number of vowels × (number of alphabets - number of vowels) ^ number of positions left to fill

7 × 5 × (21^6) = 3001814235

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