Permutations and Combinations: Formulas, Shortcuts and Solved Examples — MCQs with Answers (Set 3)

Treat the 3 vowels (AUE) of ‘DAUGHTER’ as a single block, since they must stay together. ‘DAUGHTER’ then reduces to 5 consonants plus 1 vowel-block, or 6 units to arrange. These 6 units can be arranged in 720 ways. Within the block, the 3 vowels can be arranged among themselves in 6 ways. By the counting principle, total arrangements = 720 x 6 = 4320.

Treat the 4 vowels (EAUE) of ‘MEASURE’ as a single block, since they must stay together. ‘MEASURE’ then reduces to 3 consonants plus 1 vowel-block, or 4 units to arrange. These 4 units can be arranged in 24 ways. Within the block, the 4 vowels can be arranged among themselves in 12 ways. By the counting principle, total arrangements = 24 x 12 = 288.

Treat the 3 vowels (OUI) of ‘COUSIN’ as a single block, since they must stay together. ‘COUSIN’ then reduces to 3 consonants plus 1 vowel-block, or 4 units to arrange. These 4 units can be arranged in 24 ways. Within the block, the 3 vowels can be arranged among themselves in 6 ways. By the counting principle, total arrangements = 24 x 6 = 144.

Treat the 2 vowels (OU) of ‘COUNTRY’ as a single block, since they must stay together. ‘COUNTRY’ then reduces to 5 consonants plus 1 vowel-block, or 6 units to arrange. These 6 units can be arranged in 720 ways. Within the block, the 2 vowels can be arranged among themselves in 2 ways. By the counting principle, total arrangements = 720 x 2 = 1440.

The word ‘PLANET’ has 6 letters: 2 vowels and 4 consonants. We want the 2 vowels fixed at the two end positions. The 2 vowels can be arranged in those two end spots in 2! = 2 ways. The remaining 4 consonants fill the middle positions in 4! ways. 4! = 4 x 3 x 2 x 1 = 24. By the counting principle, total arrangements = 2 x 24 = 48.

The word ‘SILVER’ has 6 letters: 2 vowels and 4 consonants. We want the 2 vowels fixed at the two end positions. The 2 vowels can be arranged in those two end spots in 2! = 2 ways. The remaining 4 consonants fill the middle positions in 4! ways. 4! = 4 x 3 x 2 x 1 = 24. By the counting principle, total arrangements = 2 x 24 = 48.

The word ‘WINTER’ has 6 letters: 2 vowels and 4 consonants. We want the 2 vowels fixed at the two end positions. The 2 vowels can be arranged in those two end spots in 2! = 2 ways. The remaining 4 consonants fill the middle positions in 4! ways. 4! = 4 x 3 x 2 x 1 = 24. By the counting principle, total arrangements = 2 x 24 = 48.

The word ‘GARDEN’ has 6 letters: 2 vowels and 4 consonants. We want the 2 vowels fixed at the two end positions. The 2 vowels can be arranged in those two end spots in 2! = 2 ways. The remaining 4 consonants fill the middle positions in 4! ways. 4! = 4 x 3 x 2 x 1 = 24. By the counting principle, total arrangements = 2 x 24 = 48.

The word ‘DESIGN’ has 6 letters: 2 vowels and 4 consonants. We want the 2 vowels fixed at the two end positions. The 2 vowels can be arranged in those two end spots in 2! = 2 ways. The remaining 4 consonants fill the middle positions in 4! ways. 4! = 4 x 3 x 2 x 1 = 24. By the counting principle, total arrangements = 2 x 24 = 48.

The word ‘MARKET’ has 6 letters: 2 vowels and 4 consonants. We want the 2 vowels fixed at the two end positions. The 2 vowels can be arranged in those two end spots in 2! = 2 ways. The remaining 4 consonants fill the middle positions in 4! ways. 4! = 4 x 3 x 2 x 1 = 24. By the counting principle, total arrangements = 2 x 24 = 48.

The word ‘COMMITTEE’ has 9 letters in total. But E occurs 2 times, M occurs 2 times, T occurs 2 times, so those repeated letters are identical to each other. Swapping identical letters gives the same arrangement, so we must divide. Number of arrangements = 9! / (2! x 2! x 2!) = 45360. So ‘COMMITTEE’ can be arranged in 45360 distinct ways.

The word ‘SUCCESS’ has 7 letters in total. But C occurs 2 times, S occurs 3 times, so those repeated letters are identical to each other. Swapping identical letters gives the same arrangement, so we must divide. Number of arrangements = 7! / (2! x 3!) = 420. So ‘SUCCESS’ can be arranged in 420 distinct ways.

The word ‘COFFEE’ has 6 letters in total. But E occurs 2 times, F occurs 2 times, so those repeated letters are identical to each other. Swapping identical letters gives the same arrangement, so we must divide. Number of arrangements = 6! / (2! x 2!) = 180. So ‘COFFEE’ can be arranged in 180 distinct ways.

The word ‘MISSISSIPPI’ has 11 letters in total. But I occurs 4 times, P occurs 2 times, S occurs 4 times, so those repeated letters are identical to each other. Swapping identical letters gives the same arrangement, so we must divide. Number of arrangements = 11! / (4! x 2! x 4!) = 34650. So ‘MISSISSIPPI’ can be arranged in 34650 distinct ways.

The word ‘BALLOON’ has 7 letters in total. But L occurs 2 times, O occurs 2 times, so those repeated letters are identical to each other. Swapping identical letters gives the same arrangement, so we must divide. Number of arrangements = 7! / (2! x 2!) = 1260. So ‘BALLOON’ can be arranged in 1260 distinct ways.

The word ‘APPLE’ has 5 letters in total. But P occurs 2 times, so those repeated letters are identical to each other. Swapping identical letters gives the same arrangement, so we must divide. Number of arrangements = 5! / (2!) = 60. So ‘APPLE’ can be arranged in 60 distinct ways.

The word ‘LETTER’ has 6 letters in total. But E occurs 2 times, T occurs 2 times, so those repeated letters are identical to each other. Swapping identical letters gives the same arrangement, so we must divide. Number of arrangements = 6! / (2! x 2!) = 180. So ‘LETTER’ can be arranged in 180 distinct ways.

The word ‘BANANA’ has 6 letters in total. But A occurs 3 times, N occurs 2 times, so those repeated letters are identical to each other. Swapping identical letters gives the same arrangement, so we must divide. Number of arrangements = 6! / (3! x 2!) = 60. So ‘BANANA’ can be arranged in 60 distinct ways.

The word ‘PLANET’ has 6 letters, and all are distinct. Since every letter is different, each position matters for the order. Number of arrangements = 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720. So ‘PLANET’ can be arranged in 720 ways.

The word ‘FIGHT’ has 5 letters, and all are distinct. Since every letter is different, each position matters for the order. Number of arrangements = 5! = 5 x 4 x 3 x 2 x 1 = 120. So ‘FIGHT’ can be arranged in 120 ways.

The word ‘BRIDGE’ has 6 letters, and all are distinct. Since every letter is different, each position matters for the order. Number of arrangements = 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720. So ‘BRIDGE’ can be arranged in 720 ways.

The word ‘WINTER’ has 6 letters, and all are distinct. Since every letter is different, each position matters for the order. Number of arrangements = 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720. So ‘WINTER’ can be arranged in 720 ways.

The word ‘TABLE’ has 5 letters, and all are distinct. Since every letter is different, each position matters for the order. Number of arrangements = 5! = 5 x 4 x 3 x 2 x 1 = 120. So ‘TABLE’ can be arranged in 120 ways.

The word ‘PENCIL’ has 6 letters, and all are distinct. Since every letter is different, each position matters for the order. Number of arrangements = 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720. So ‘PENCIL’ can be arranged in 720 ways.

The word ‘MOBILE’ has 6 letters, and all are distinct. Since every letter is different, each position matters for the order. Number of arrangements = 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720. So ‘MOBILE’ can be arranged in 720 ways.