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# Scratch Cards

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Age 14 to 16

Challenge Level

- Problem
- Student Solutions

Here is another nut that has been waiting to be cracked for months. Three solutions arrived simultaneously from Bradley, Avery Coonley School, USA; Joe, Anglo Chinese School, Singapore and Ling Xiang Ning, Allan, Tao Nan School, Singapore, whose solution is given below.

Mail order companies often send a scratch card with their catalogue. These cards are used to encourage people to order from the catalogue by offering a free price. You have to uncover three of the numbers by scratching off the cover and revealing the number under three of the circles. Of course the numbers under the circles are randomly placed. If the numbers add up to more than fifteen you win a prize. What is the probability of winning?As there are 9*8*7 possible choices of 3 numbers, and the order of revealing the numbers does not matter, the number of choices of the three numbers is (9*8*7)/6 = 84

The probability of winning is the number of winning combinations divided by 84.

We have to count the number of winning combinations:

We start from 987, 986? and the first number chosen gets smaller, and the other numbers must be smaller than the first so that three numbers will not appear together more than once.

There are total 38 choices, as shown:

987, 986, 985, 984, 983, 982, 981, 976, 975, 974, 973, 972, 971,
965, 964, 963, 962, 961, 954,

953, 952, 943, 876, 875, 874, 873, 872, 871, 865, 864, 863, 862,
854, 853, 765, 764, 763, 754

So, the probability of winning a price is 38/84 or also, 19/42.