Let ( array ll n k array ) denote Ck n and [ array ll n k array ] =...

${A}_{k} = \mathop{∑}_{i = 0}^{9}{9}_{{C}_{i}} \cdot {C}_{k - i} 12 + \mathop{∑}_{i = 0}^{8}{8}_{{C}_{i}} \cdot {C}_{k - i} 13$

=Coefficient of ${x}^{k}$ in ${\left(1 + x\right)}^{9} \times {\left(1 + x\right)}^{12} +$Coefficient of ${x}^{k}$ in ${\left(1 + x\right)}^{8} \times {\left(1 + x\right)}^{13}$

=Coefficient of ${x}^{k}$ in ${\left(1 + x\right)}^{21} +$ Coefficient of ${x}^{k}$ in ${\left(1 + x\right)}^{21}$

${A}_{k} = Ck 21 + Ck 21 = 2 . Ck 21$

${A}_{4} = \mathop{∑}_{i = 0}^{9}{9}_{{C}_{i}} \cdot {C}_{4 - i} 12 + \mathop{∑}_{i = 0}^{8}{8}_{{C}_{i}} \cdot {C}_{4 - i} 13$

$= C4 21 + C4 21 = 2\left(C4 21\right)$

${A}_{3} = C3 21 + C3 21 = 2\left(C3 21\right)$

${A}_{4} - {A}_{3} = 2\left(C4 21 - C3 21\right) = 190p$

$⇒ p = 49$