Question

A purse contains four fifty-paise coins, three two-rupees coins and three five-rupee coins. If three coins are selected at random, then what is the probability of getting the minimum amount? (1) \(1 / 15\) (2) \(1 / 10\) (3) \(1 / 30\) (4) \(1 / 5\)

Short answer

Answer: The probability is 1/30.

Step-by-step solution

Calculate the total number of ways to select three coins

There are a total of 4 fifty-paise coins, 3 two-rupee coins, and 3 five-rupee coins, making a total of 10 coins. We need to select 3 coins from these 10. We can use the formula for combinations to calculate this: Number of ways to select 3 coins from 10 = \(\binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10!}{3!7!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120\) ways

Calculate the number of ways to select all three fifty-paise coins

There are 4 fifty-paise coins, and we need to select all 3 of them. So, we can use the formula for combinations again: Number of ways to select all three fifty-paise coins = \(\binom{4}{3} = \frac{4!}{3!(4-3)!} = \frac{4!}{3!1!} = \frac{4}{1} = 4\) ways

Calculate the probability of selecting the three fifty-paise coins

Now, we'll divide the number of ways to select all three fifty-paise coins by the total number of ways to select three coins: Probability = \(\frac{\text{Number of ways to select all three fifty-paise coins}}{\text{Total number of ways to select three coins}} = \frac{4}{120} = \frac{1}{30}\) The probability of selecting the three fifty-paise coins and thus getting the minimum amount is \(\frac{1}{30}\). Therefore, the correct answer is option (3) \(1 / 30\).

Key concepts

Combinations in Probability

Understanding combinations in probability is like learning the recipe for a mathematical cake. Imagine you have a box of different colored beads and you want to create a bracelet by choosing a few at a time. Combinations allow us to calculate how many different bracelets we can make based on our selection. This is essential in probability as it helps us find out the number of ways events can happen.

For example, in our exercise, we have coins of different denominations and we need to determine how many ways we can pick three coins. The combination formula that applies here is written as \( \binom{n}{r} \) where \( n \) is the total number of items to choose from and \( r \) is the number of items to choose. It’s calculated using the formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). Factorials, represented by the exclamation mark \( ! \), play a crucial role here. We will dive into factorials in the next section for a clearer picture.

Factorial Notation in Mathematics

The concept of factorial notation in mathematics may sound complex, but it's actually quite a simple concept once you break it down. Think of it like watching a domino effect; one piece knocks over the next, and this continues until there are no more pieces to fall over. A factorial is similarly represented by a number followed by an exclamation point, like this: \( 5! \), which means you multiply the number by each positive integer less than itself. So, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

Back to our coin example, when we're looking for ways to select three fifty-paise coins from four (\( \binom{4}{3} \) ), we calculate \( 4! \) for the numerator and \( 3! \times 1! \) for the denominator. The factorials help us to ensure that we count all possible selections without repetition, which is fundamental in combination problems in probability.

Probability Calculation Methods

Exploring probability calculation methods is akin to navigating a treasure map. There are various routes you can take to find the treasure, or in our case, the likelihood of an event occurring. We've already seen one such method involving combinations. However, there are also other techniques that include addition and multiplication rules, conditional probability, and the use of probability trees to visualize complex events.

Our focus here, though, is on using combinations to calculate the probability of an event. This involves dividing the favorable outcomes (selecting three fifty-paise coins) by all possible outcomes (selecting any three coins from ten). As we've calculated, there are 4 ways to select the fifty-paise coins and 120 ways to select any three coins. Thus, the probability is \( \frac{4}{120} = \frac{1}{30} \), which allows us to arrive at the solution for our given exercise.