First slide

Combinations

Question

There are 3 set of parallel lines containing respectively $p$ lines, $q$ lines and $r$ lines respectively. The greatest number of parallelograms that can be formed by the system

Moderate

A

$p q r + (p - 1) (q - 1) (r - 1)$
B

$\frac{1}{4} {p q r + (p - 1) (q - 1) (r - 1)}$
C

$\frac{1}{4} pqr (p + 1) (q + 1) (r + 1)$
D

none of these

Solution

Number of parallelograms
$= (^{p} C_{2}) (^{q} C_{2}) + (^{q} C_{2}) (^{r} C_{2}) + (^{r} C_{2}) (^{p} C_{2})$

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