Question: A marine conservation initiative tags 10 sea turtles and 15 coral fragments. If four items are randomly selected, what is the probability that at least one sea turtle is chosen?

February 22, 2026

Question: A marine conservation initiative tags 10 sea turtles and 15 coral fragments. If four items are randomly selected, what is the probability that at least one sea turtle is chosen?

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Solution: First, choose 4 positions out of 10 for the turtle shells, and the remaining 6 will hold starfish. The number of arrangements is $\binom{10}{4} = 210$. Hence, the answer is $\boxed{210}$.
Question: A box contains 7 green educational badges and 9 digital certificates. If three items are selected at random, what is the probability that exactly 2 are badges?
Solution: The total number of ways to choose 3 items is $\binom{16}{3} = 560$. The favorable cases are $\binom{7}{2} \times \binom{9}{1} = 21 \times 9 = 189$. The probability is $\frac{189}{560} = \frac{27}{80}$. Thus, the answer is $\boxed{\dfrac{27}{80}}$.
Solution: The total ways to select 4 items: $\binom{25}{4} = 12650$. The unfavorable case (no turtles) is $\binom{15}{4} = 1365$. The probability of at least one turtle is $1 - \frac{1365}{12650} = 1 - \frac{273}{2530} = \frac{2257}{2530}$. The answer is $\boxed{\dfrac{2257}{2530}}$.
Question: A student designs a bridge with 5 red struts and 7 blue connectors. If all struts and connectors are placed in a line, how many distinct sequences are possible if struts of the same color are indistinct?
Solution: The total items are $5 + 7 = 12$, with 5 red and 7 blue. The number of distinct sequences is $\frac{12!}{5!7!} = 792$. Therefore, the answer is $\boxed{792}$.Question: A robotic engineer is programming a machine to assemble components using exactly 3 different colored

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