Two distinct points are chosen randomly out of the points $A_1, A_2, \ldots, A_{49}$. Let $p$ be the probability that they are friends. Then the value of $7p$ is

$\text{Points } A_1, A_2, \ldots, A_{49} \text{ are given}$

$2 \text{ distinct points chosen at random (unordered)}$

$p = P(\text{the two chosen points are friends})$

$\text{To find: } 7p$

$A_i, A_j \text{ are friends} \iff i + j \equiv 0 \pmod 7$

$\text{Total outcomes} = \binom{49}{2} = \dfrac{49 \times 48}{2} = 1176$

$\text{Residues of } 1, \ldots, 49 \pmod 7:\ \text{each of } 0, 1, \ldots, 6 \text{ appears } 7 \text{ times}$

$\text{Both} \equiv 0 \pmod 7:\quad \binom{7}{2} = 21$

$\text{Opposite residues } \{a, 7-a\}:\quad 3 \times (7 \times 7) = 147$

$\text{Friendly pairs} = 21 + 147 = 168$

$p = \dfrac{168}{1176} = \dfrac{1}{7}$

$7p = 7 \times \dfrac{1}{7} = 1$