1980 AMC 12 Problem 20

Question

A box contains

2

pennies,

4

nickels and

6

dimes. Six coins are drawn without replacement, with each coin having an equal probability of being chosen. What is the probability that the value of the coins drawn is at least

50

cents?

\frac{37}{924}

\frac{91}{924}

\frac{127}{924}

\frac{132}{924}

none of these

Answer: C

Concepts:combinations sampling without replacement casework

Difficulty rating: 2100

Small hint:

There are

\binom{12}{6}

equally likely sets of six coins

Big hint:

Classify favorable selections by whether they contain six, five, or four dimes

Solution:

There are

\binom{12}{6}=924

selections. A value of at least

50

cents occurs with six dimes; with five dimes and any one other coin; or with four dimes and two nickels. The favorable count is

\begin{aligned} &\binom66\\ &\quad+\binom65\binom61\\ &\quad+\binom64\binom42\\ &=1+36+90\\ &=127. \end{aligned}

Thus the probability is

\frac{127}{924}.

Therefore, the correct answer is C.