# Permutation Problems - 01

**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**

**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**

**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**

**Problem 4**

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

**Solution 4**

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**

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**

Number of alike consonants

Number of letter T = 2

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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