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

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 (A) p q r+(p-1)(q-1)(r-1) (B) (1/4) p q r+(p-1)(q-1)(r-1)  (C) (1/4) operatornamepqr(p+1)(q+1)(r+1) (D) none of these

Tardigrade Admin · Accepted Answer

Number of parallelograms =( p C2)( q C2)+( q C2)( r C2)+( r C2)( p C2)

Q. 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

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

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

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

none of these

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

Q. 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

pqr+(p−1)(q−1)(r−1)

41​{pqr+(p−1)(q−1)(r−1)}

41​pqr(p+1)(q+1)(r+1)