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Mathematics-Online problems:

Solution to the problem of the (previous) week


Problem:

A sports team consisting of 4 girls and 4 boys, is asked to form groups of two kids.

Lara     Marc
OLGA     Axel
Vera     TIMO
Elke     Hans


How many possibilities exist, if

a) there are only girls or only boys in each group,
b) each group has one girl and one boy,
c) there is no condition at all?


Answer:

a)    
b)    
c)

   


Solution:

a)
There exist 3 possibilities for the two groups of girls. This is, e.g., determined by the fact which of the girls forms a group with Lara. If we combined them with the corresponding possibilities for the grouping of the boys, we would obtain

$\displaystyle 3^2 = 9
$

possibilities.


b)
We consecutively count the possibilities of choosing a partner for Lara, Olga, Vera, and Elke. Thus, we obtain

$\displaystyle 4 \cdot 3 \cdot 2 \cdot 1 = 24
$

possibilities.


c)
There are

$\displaystyle {n \choose 2} = \frac{n \cdot(n-1)}{2}
$

possibilities to group $ n$ persons in pairs. If we successively made groups of two from the 8 children and considered that the sequence can be neglected, we would have

$\displaystyle {8 \choose 2} \cdot {6 \choose 2} \cdot {4 \choose 2}\, / \,\, 4! = 105
$

possibilities.


[problem of the week]