Problem 96 - Algebra for JEE ADVANCED

Question

The total number of ways of selecting six coins out of 20 onerupee coins, 10 fifty-paise coins and 7 twenty-five paise coins is a. 28 b. 56 c \(\quad{ }^{37} \mathrm{C}_{6}\) d. none of these

Step-by-Step Solution

Answer

The answer is c \( ^{37} \mathrm{C}_{6} \).

Step 1: Understanding the Problem

We are given 20 one-rupee coins, 10 fifty-paise coins, and 7 twenty-five paise coins. We need to find the total number of ways to select 6 coins from these 37 coins (20 + 10 + 7). We are choosing without replacement, and the order in which we choose doesn't matter.

Step 2: Applying Combination Formula

The problem requires us to find the combination of choosing 6 coins from a total of 37 coins. We'll use the combination formula given by \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \) where \( n \) is the total number of items and \( r \) is the number of items to choose.

Step 3: Setup the Formula

Let's set up the formula to calculate \( \binom{37}{6} \). Here, \( n = 37 \) and \( r = 6 \). So, the expression becomes: \[ \binom{37}{6} = \frac{37!}{6!(37-6)!} = \frac{37!}{6! \times 31!} \]

Step 4: Calculate the Combination

To calculate \( \binom{37}{6} \), we can simplify the factorial expression:\[ \frac{37 \times 36 \times 35 \times 34 \times 33 \times 32}{6 \times 5 \times 4 \times 3 \times 2 \times 1} \]. Calculating this gives:1. \( 37 \times 36 = 1332 \)2. \( 1332 \times 35 = 46620 \)3. \( 46620 \times 34 = 1585080 \)4. \( 1585080 \times 33 = 52207640 \)5. \( 52207640 \times 32 = 1670644480 \)6. Divide by \( 720 \) (which is \( 6! \)) gives 435897.

Step 5: Conclude the Solution

The calculated value from the formula corresponds to \( \binom{37}{6} \). Therefore, the total number of ways of selecting six coins out of the given lots is \( \binom{37}{6} \), which matches option c.

Key Concepts

Combination Formula

When we talk about combinations in mathematics, we are discussing the different ways we can choose items from a group, where order does not matter. For instance, picking coins from a collection of coins is a typical example of using a combination.

• The formula used to calculate combinations is denoted as \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \).
• Here, \( n \) represents the total number of items, and \( r \) is the number of items you want to select.

In these expressions, \( \binom{n}{r} \) is often read as "n choose r". It's crucial to understand that factoring in order would result in permutations, not combinations.
The combination formula is incredibly useful in scenarios like lottery odds, creating teams, or any situation where selection without regard to sequence is needed.

Factorial Calculation

Factorial calculations are foundational when working with permutations and combinations. A factorial is depicted by an exclamation mark (!). It represents the product of all positive integers up to a certain number.

• For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
• In the combination formula, factorials are used to account for numerous elimination possibilities, as in \( r!(n-r)! \).

This might seem complex at first, but remember, each number increments and reduces multiplication to a simpler form.
When computing combinations like \( \binom{37}{6} \), the use of factorials simplifies the vast multiplication into manageable components that cancel each other out in the denominator.
Factorials grow rapidly, which shows their powerful capacity to calculate larger combinations without listing each possibility.

Selection Without Replacement

Selection without replacement means once an item is chosen, it is not put back into the pool of things you can choose from next. This method applies to the exercise where we pick coins.

• This concept critically affects the probabilities as it decreases the number of available options gradually.
• For example, if you pick one coin from a pile, and it is not returned, your next pick will be from a smaller assortment.

This is different from scenarios with replacement, where every pick has the same number of choices. Without replacement, the situation represents real-world conditions like lottery draws or drawing cards where once used, an item cannot be selected again.
Handling selections in this manner ensures the final count reflects true combinations, adjusting the options accurately as each item is chosen.