Alice has a shuffled deck of 11 cards that have numbers 15,32, 6, 17,22, 43, 67,

Question

Alice has a shuffled deck of 11 cards that have numbers 15,32, 6, 17,22, 43, 67, 82, 11,99,97). She chooses 2 cards out of these 11 cards. Note that the order with which Alice chooses the two cards does not matter (a) Find the total number of possible choices of two cards (b) Find the number of choices for which both cards have an even number (c) Find the number of choices for which both cards have a prime number (d) Find the number of choices for which there is one card that has even number and the other has an odd number (e) Find the number of choices for which at least one card has an even number

Explanation / Answer

nCx = n! / [(n-x)!*x!]

Arranging the numbers we get 4 even numbers 6, 32, 22, 82. 7 odd numbers of which 5 are prime (highlighted)

15, 17, 43, 67, 11, 97, 99

(a) Total number of possible choices of 2 cards = 11C2 = 11! / [(9)!*2!] = (11 * 10)/2 = 55

(b) Total number of possible choices of 2 cards where both are even = 4C2 = 4! / [(2)!*2!] = (4 * 3)/2 = 6

(c) Total number of possible choices of 2 cards where both are prime = 5C2 = 5! / [(3)!*2!] = (5 * 4)/2 = 10

(d) Total number of possible choices of 2 cards where 1 is even and the other odd = 4C1 * 7C1 = 4 * 7 = 28

(e) Total number of possible choices of 2 cards where at least 1 is even.

At least 1 means (1 even and 1 odd) + (Both even) = 28 + 6 = 34

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