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Exercise 1
In how many ways can the letters of the word 'APPLE' be arranged ?
A. 720 B. 120 C. 60 D. 180 Answer & Explanation
Answer: Option C
Explanation:
The word 'APPLE' contains 5 letters, 1A, 2P, 1L.and 1E.
$$\therefore$$ Required number of ways = $$\frac{5 !}{[1 !] [2 !] [1 !] [1 !]}$$ = 60.
How many 4-letter words with or without meaning, can be formed out of the letters of the word, 'LOGARITHMS', if repetition of letters is not allowed ?
A. 40 B. 400 C. 5040 D. 2502 Answer & Explanation
Answer: Option C
Explanation:
'LOGARITHM' contains 10 different letters.
Required number of words = Number of arrangements of 10 letters, taking 4 at a time = 10P4 = [10 * 9 * 8 * 7] = 5040.
The value of 75P2 is :
A. 2775 B. 150 C. 5550 D. None of these Answer & Explanation
Answer: Option C
Explanation: 75P2 = $$\frac{75 !}{[75 - 2]!}$$ = $$\frac{75 !}{73 !}$$ = $$\frac{75 * 74 * [73 !]}{73 !}$$ = [75 * 74] = 5550.
In how many ways can the letters of the word 'LEADER' be arranged ?
A. 72 B. 144 C. 360 D. 720 Answer & Explanation
Answer: Option C
Explanation:
The word 'LEADER' contains 6 letters, namely 1L, 2E, 1A, 1D and 1R.
$$\therefore$$ Required number of ways = $$\frac{6 !}{[1 !][2 !][1 !][1 !][2!]}$$ = 360.
How many words with or without meaning, can be formed by using all the letters of the word, 'DELHI', using each letter exactly once ?
A. 10 B. 25 C. 60 D. 120 Answer & Explanation
Answer: Option D
Explanation:
The word 'DELHI' contains 5 different letters.
Required number of words = Number of arrangements of 5 letters, taken all at a time = 5P5 = 5 ! = [5 *4 *3 *2 *1] = 120.
In how many different ways can the letters of the word 'RUMOUR' be arranged ?
A. 180 B. 90 C. 30 D. 720 Answer & Explanation
Answer: Option A
Explanation:
The word 'RUMOUR' contains 6 letters, namely 2R, 2U, 1M and 1U.
$$\therefore$$ Required number of ways = $$\frac{6 !}{[2 !] [2 !] [1 !] [1!]}$$ = 180.
How many arrangements can be made out of the letters of the word 'ENGINEERING' ?
A. 277200 B. 92400 C. 69300 D. 23100 Answer & Explanation
Answer: Option A
Explanation:
The word 'ENGINEERING' contains 11 letters, namely 3E, 3N, 2G, 2I and 1R.
$$\therefore$$ Required number of arrangements = $$\frac{11 !}{[3 !] [3 !] [2 !][2 !][1 !]}$$ = 277200.
How many words can be formed from the letters of the word 'SIGNATURE' so that the vowels always come together ?
A. 720 B. 1440 C. 2880 D. 17280 Answer & Explanation
Answer: Option D
Explanation:
The word 'SIGNATURE' contains 9 different letters.
When the vowels IAUE are taken together, they can be supposed to form an entity, treated as one letter.
Then, the letters to be arranged are SGNTR [IAUE].
These 6 letters can be arranged in 6P6 = 6 ! = 720 ways.
The vowels in the group [IAUE] can be arranged amongst themselves in 4P4 = 4 ! = 24 ways.
$$\therefore$$ Required number of words = [720 * 24] = 17280.
In how many different ways can the letters of the word 'SOFTWARE' be arranged in such a way that the vowels always come together ?
A. 120 B. 360 C. 1440 D. 720 Answer & Explanation
Answer: Option D
Explanation:
The word 'SOFTWARE' contains 8 different letters.
When the vowels OAE are always together, they can be supposed to form one letter.
Thus, we hdve to arrange the letters SFTWR [OAE].
Now, 5 letters can be arranged in 6 ! = 720 ways.
The vowels [OAE] can be arranged among themselves in 3 ! = 6 ways.
$$\therefore$$ Required number of ways = [720 * 6] = 4320.
In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels ajways come together ?
A. 120 B. 720 C. 4320 D. 2160 Answer & Explanation
Answer: Option C
Explanation:
The word 'OPTICAL' contains 7 different letters.
When the vowels OIA are always together, they can be supposed to form one letter.
Then, we have to arrange the letters PTCL [OIA].
Now, 5 letters can be arranged in 5 ! = 120 ways.
The vowels [OIA] can be arranged among themselves in 3 ! = 6 ways.
$$\therefore$$ Required number of ways = [120 * 6] = 720.