Problem 1 In the 1500-m race in Olympics, fourteen top tier athletes compete. How many ways can the gold, silver and bronze medals be awarded?

Solution 1

Solution by Permutation $N = {^{14}}P_3 = 2184 ~ \text{ways}$

Problem 2 How many ways can 8 cadets stand in a row?

Solution 2

$N = 8! = 40,320 ~ \text{ways}$

Solution by Permutation $N = {^8}P_8 = 40,320 ~ \text{ways}$

Problem 3 In how many ways can the letters of the word THANOS be arranged?

Solution 3

Problem 4 In how many ways can the letters of the word MATHALINO be arranged?

Solution 4

The permutation of 9 objects where 2 are alike is $N = \dfrac{9!}{2!} = 181,440 ~ \text{ways}$

Problem 5 In how many ways can the letters of the word MATHALINO be arranged if the vowels are to come together?

Solution 5

Consider the vowels as one object so that there are 6 objects to be arranged, namely; M, T, H, L, N, AAIO. Note that AAIO can be arranged within their group. $N = 6! \times \dfrac{4!}{2!} = 8,640 ~ \text{ways}$

Problem 6 In how many ways can the letters of the word MATHEMATICS be arranged if the consonants are to come together?

Solution 6

Number of alike consonants

Number of vowels = 4

Number of alike vowels

Consider the consonants as one object so that there are 5 objects to be arranged, namely; A, E, A, I, MTHMTCS. Note that MTHMTCS can be arranged within their group. $N = \dfrac{5!}{2!} \times \dfrac{7!}{2! \times 2!} = 75,600 ~ \text{ways}$

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