Question

For a game in which two partners play against two other partners, six persons are available. If every possible pair must play every other possible pair, then the total number of games played is
A) 90
B) 45
C) 30
D) 60

Answer 1

Hint: A combination is a mathematical technique for determining the number of possible arrangements in a set of items where the order of the selection is irrelevant. You can choose the items in any order in combinations.

Complete step-by-step solution:
The order of the digits was important in our example; if the order did not matter, we would have what is the definition of a combination. The following formula determines the number of combinations of n objects taken r at a time: \[C\left( n,r \right){{=}^{n}}{{\text{C}}_{r}}=\dfrac{n!}{\left( n-r \right)!r!}\]
\[\Rightarrow ^{n}{{\text{C}}_{r}}=\] Number of combinations
\[\Rightarrow n\]=total number of objects in the set
\[\Rightarrow r\]=number of choosing objects from the set
Here, Four people are required for one game.
This can be done in \[{}^{6}{{C}_{4}}=15~ways\]
once a set of four persons are selected, number of games possible will be
\[\text{ }C\left( n,r \right){{=}^{n}}{{\text{C}}_{r}}=\dfrac{n!}{\left( n-r \right)!r!}\] games.
Therefore the total number of possible will be = \[3\times 15=45.\]
Thus the correct answer is option ‘B’.
Note: Combinations are the taking of n things k at a time without repetition. The terms k-selection, k-multiset, and k-combination with repetition are frequently used to refer to combinations in which repetition is permitted. If it were possible to have two of any one type of fruit in the preceding example, there would be three more 2-selections: one with two apples, one with two oranges, and one with two pears.