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Largest numbered ticket

ProbabilityMedium

An urn holds $10$ tickets numbered $1$ to $10$ . You draw $3$ tickets at random without replacement. What is the probability that the largest number drawn is exactly $7$ ? Give a fraction in lowest terms.

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Hint

Fix the ticket numbered $7$ and choose the other tickets only from smaller numbers.

Solution

The number of equally likely sets of $3$ tickets is $(3 10) = 120$ .
For the largest to be $7$ , the ticket $7$ must be drawn and the other $2$ tickets come from the numbers $1$ to $6$ .
This can be done in $(2 6) = 15$ ways.
The probability is $\frac{15}{120}$ , which reduces to $\frac{1}{8}$ .

Answer: $\frac{1}{8}$

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