six lettered tiles are drawn from a bag and placed in a row .if 3 tiles have 3 x's and 3 have o's.find the probability that no 2 adjacent tiles have the same letter?
To find the probability that no two adjacent tiles have the same letter, we need to first calculate the total number of possible arrangements and then determine the number of favorable arrangements where no two adjacent tiles have the same letter.
Step 1: Calculate the total number of possible arrangements
Since there are 6 tiles in total, the number of possible arrangements is given by 6 factorial (6!):
Total arrangements = 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720
Step 2: Calculate the number of favorable arrangements
To determine the number of favorable arrangements, we need to consider the possible patterns where no two adjacent tiles have the same letter.
Pattern 1: XOXOXO
Pattern 2: OXOXOX
For each pattern, we can arrange the three X's and three O's within that pattern. Let's calculate the number of arrangements for each pattern.
Pattern 1: XOXOXO
1. Arrange the X's: There are 3 X's to arrange, which can be done in 3! ways.
2. Arrange the O's: There are 3 O's to arrange, which can also be done in 3! ways.
Number of arrangements for Pattern 1 = 3! x 3! = 6 x 6 = 36
Pattern 2: OXOXOX
1. Arrange the X's: There are 3 X's to arrange, which can be done in 3! ways.
2. Arrange the O's: There are 3 O's to arrange, which can also be done in 3! ways.
Number of arrangements for Pattern 2 = 3! x 3! = 6 x 6 = 36
Step 3: Calculate the probability
The probability is calculated by dividing the number of favorable arrangements by the total number of possible arrangements.
Probability = Number of favorable arrangements / Total number of possible arrangements
Probability = (Number of arrangements for Pattern 1 + Number of arrangements for Pattern 2) / Total arrangements
Probability = (36 + 36) / 720
Probability = 72 / 720
Probability = 1 / 10
Probability = 0.1
Therefore, the probability that no two adjacent tiles have the same letter is 0.1 or 1/10.