Permutation Problems - 01 | Algebra Review at MATHalino

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

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Slot Method
$N = 14 \times 13 \times 12 = 2184 ~ \text{ways}$

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

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

Solution 2

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Slot Method
$N = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$

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

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

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$N = 6! = 720 ~ \text{ways}$

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

Solution 4

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Number of letters = 9
Number of letter A = 2

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

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Number of consonants = 5
Number of vowels = 4
Number of alike vowels = 2

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

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Number of consonants = 7

Number of alike consonants

Number of letter M = 2
Number of letter T = 2

Number of vowels = 4

Number of alike vowels

Number of letter A = 2

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.
$N = \dfrac{5!}{2!} \times \dfrac{7!}{2! \times 2!} = 75,600 ~ \text{ways}$

Tags:

Permutation

Olympics Race

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