If the number of ways in which a mixed double badminton can be played such that no couples played into a same game is 840. Then find the number of players?
Approach Solution - 1
Let the number of couples be \( n \).
The total number of ways is given by:
\( \frac{n(n-1)}{2} \cdot \frac{(n-2)(n-3)}{2} \cdot 2 = 840 \).
Step 1: Simplify the equation:
\( n(n-1)(n-2)(n-3) = 840 \cdot 4 \).
\( n(n-1)(n-2)(n-3) = 3360 \).
Step 2: Factorize \( 3360 \):
\( 3360 = 8 \cdot 7 \cdot 6 \cdot 5 \).
Thus, \( n = 8 \).
Step 3: Calculate the total number of persons:
Total persons = \( 2n = 2(8) = 16 \).
Final Answer: The number of persons is \( 16 \).
Approach Solution -2
According to given condition,
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Concepts Used:
Permutations and Combinations
Permutation:
Permutation is the method or the act of arranging members of a set into an order or a sequence.
- In the process of rearranging the numbers, subsets of sets are created to determine all possible arrangement sequences of a single data point.
- A permutation is used in many events of daily life. It is used for a list of data where the data order matters.
Combination:
Combination is the method of forming subsets by selecting data from a larger set in a way that the selection order does not matter.
- Combination refers to the combination of about n things taken k at a time without any repetition.
- The combination is used for a group of data where the order of data does not matter.