how many ways can 6 coins be chosen from 20, one rupee coins, 10 fifty paise coins, 7 twenty paise coins?
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To calculate the number of ways to choose 6 coins from the given set, we can use the concept of combinations.
First, let's consider the different types of coins we have: one rupee coins, fifty paise coins, and twenty paise coins. We need to choose 6 coins from these three types.
To determine the number of ways, we'll break it down into three cases based on the types of coins chosen:
Case 1: Selecting only one rupee coins:
In this case, we need to choose 6 coins from the available one rupee coins. Since we have 20 one rupee coins, we can select 6 of them in C(20, 6) ways, where C(n, r) represents the combination of n objects taken r at a time. This can be calculated using the formula C(n, r) = n! / (r!(n-r)!), where "!" denotes the factorial operation.
Case 2: Selecting only fifty paise coins:
In this case, we need to choose 6 coins from the available fifty paise coins. Since we have 10 fifty paise coins, we can select 6 of them in C(10, 6) ways.
Case 3: Selecting both fifty paise coins and twenty paise coins:
In this case, we need to choose a combination of fifty paise coins and twenty paise coins that sum up to 6 coins. We can iterate through different possibilities and calculate their combinations.
Let's break down this case even further to make it easier to calculate:
- Selecting 0 fifty paise coins and 6 twenty paise coins: C(7, 6) ways.
- Selecting 1 fifty paise coin and 5 twenty paise coins: C(10, 1) * C(7, 5) ways.
- Selecting 2 fifty paise coins and 4 twenty paise coins: C(10, 2) * C(7, 4) ways.
- Selecting 3 fifty paise coins and 3 twenty paise coins: C(10, 3) * C(7, 3) ways.
- Selecting 4 fifty paise coins and 2 twenty paise coins: C(10, 4) * C(7, 2) ways.
- Selecting 5 fifty paise coins and 1 twenty paise coin: C(10, 5) * C(7, 1) ways.
- Selecting 6 fifty paise coins and 0 twenty paise coins: C(10, 6) ways.
Finally, to get the total number of ways, we sum up the results from all three cases:
Total number of ways = C(20, 6) + C(10, 6) + [Case 3 possibilities]
You can use a calculator or a programming language that supports factorial calculations and combinations to compute the final result.