Let's solve the problem step by step.
1. Determining Total Number of Sequences:
- The child can choose from 4 different colors: red, yellow, blue, and purple.
- The sequence is made up of 3 positions.
- Since once a color is used, it cannot be repeated, we are dealing with permutations of 4 colors taken 3 at a time.
- The number of possible sequences is calculated as [tex]\(P(4,3)\)[/tex], which is the permutations of 4 items taken 3 at a time.
- The formula for permutations is given by [tex]\[ P(n, r) = \frac{n!}{(n - r)!} \][/tex]
- Plugging in the values: [tex]\( n = 4 \)[/tex] and [tex]\( r = 3 \)[/tex], we get:
[tex]\[
P(4, 3) = \frac{4!}{(4 - 3)!} = \frac{4!}{1!} = \frac{24}{1} = 24
\][/tex]
2. Calculating the Probability of Guessing the Correct Sequence on the First Try:
- There is only 1 correct sequence out of these 24 possible sequences.
- The probability [tex]\( P \)[/tex] of guessing the correct sequence on the first try is therefore:
[tex]\[
P = \frac{1}{24}
\][/tex]
Thus, the probability that the sequence is guessed on the first try is [tex]\( \frac{1}{24} \)[/tex].
Answer: [tex]\( \frac{1}{24} \)[/tex]
