jmc

Question

Forty cards are placed into a box, each bearing a number 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10, with each number entered on four cards. Four cards are drawn from the box at random and without replacement. Let p be the probability that all four cards bear the same number. Let q be the probability that three of the cards bear a number a and the other bears a number b that is not equal to a. What is the value of q/p?

Accepted Answer

The total number of ways that the numbers can be chosen is

(4 40) .

Exactly 10 of these possibilities result in the four cards having the same number.

Now we need to determine the number of ways that three cards can have a number

a

and the other card have a number

b

, with

b \neq = a

. There are

10 \cdot 9 = 90

ways to choose the distinct numbers

a

and

b

. (Notice that the order in which we choose these two number matters, since we get 3 of

a

and 1 of

b

.)

For each value of

a

there are

(3 4)

ways to choose the three cards with

a

and for each value of

b

there are

(1 4)

ways to choose the card with

b

. Hence the number of ways that three cards have some number

a

and the other card has some distinct number

b

is

90 \cdot (3 4) \cdot (1 4) = 90 \cdot 4 \cdot 4 = 1440.

So the probabilities

p

and

q

are

\frac{10}{( 4 40 )}

and

\frac{1440}{( 4 40 )}

, respectively, which implies that

\frac{q}{p} = \frac{1440}{10} = 144 .