Question

in a state lottery, there are 14 finalists who are eligible for the big money draw. in how many ways can the first, second, and third prizes be awarded if no ticket holder can win more than one prize?
ways

resources

1. -/2 points

(a) how many seven - digit telephone numbers are possible if the first digit must be non - zero?
possible numbers

(b) how many international direct - dialing numbers for calls within the united states and canada are possible if each number consists of a 1 plus a three - digit area code (the first digit of which must be non - zero) and a number of the type described in part (a)?
possible numbers

Explanation:

Step1: Calculate lottery prize permutations

We use permutations since order (1st, 2nd, 3rd) matters and no repeats: $P(n,k)=\frac{n!}{(n-k)!}$ where $n=14$, $k=3$.
$$P(14,3)=14\times13\times12$$

Step2: Calculate 7-digit phone numbers (a)

First digit: 9 options (1-9), remaining 6 digits: 10 options each.
$$9\times10^6$$

Step3: Calculate international numbers (b)

Area code: 9×10×10 options, plus 1 prefix and part (a) number.
$$1\times(9\times10^2)\times(9\times10^6)$$

Answer:

Lottery prize ways: 2184
(a) Possible seven-digit numbers: 9000000
(b) Possible international numbers: 8100000000