To determine the probability of guessing the correct sequence on the first try, let's go through the problem step by step.
1. Identify the total number of colors and sequence length:
- The available colors are: red, yellow, blue, and purple.
- The sequence length is 3.
2. Calculate the total number of possible sequences:
Since each color can only be used once and the sequence matters (the order is crucial), this is a problem of permutations.
The number of permutations of [tex]\( n \)[/tex] items taken [tex]\( k \)[/tex] at a time is given by:
[tex]\[
P(n, k) = \frac{n!}{(n - k)!}
\][/tex]
Here, [tex]\( n = 4 \)[/tex] (the number of colors) and [tex]\( k = 3 \)[/tex] (the length of the sequence).
We need to find [tex]\( P(4, 3) \)[/tex]:
[tex]\[
P(4, 3) = \frac{4!}{(4 - 3)!} = \frac{4!}{1!} = \frac{4 \times 3 \times 2 \times 1}{1} = 4 \times 3 \times 2 = 24
\][/tex]
3. Determine the probability:
The probability of guessing the sequence correctly on the first try is the reciprocal of the number of possible sequences.
[tex]\[
\text{Probability} = \frac{1}{P(4, 3)} = \frac{1}{24}
\][/tex]
Thus, the probability that the sequence is guessed correctly on the first try is:
[tex]\[
\boxed{\frac{1}{24}}
\][/tex]
This corresponds to the first choice provided in the given options.
