Let $A=\{1,2,3,...,100\}$. In how many ways $a_1\lt a_2\lt a_3\lt a_4\lt a_5, a_i \in A$ can be chosen from $A$ such that $a_p -a_i \geqslant 2$ for $(p-i)=1$
My try: From $A$ we can choise 5 distinct element by ${100 \choose 5}$ ways. If distance of two $a_i$ is exactly 1 then first element can be chosen by ${96 \choose 1}$ ways and rest 4 can be chosen by the consecutive of one another. So the way to pick 5 elements such that $a_p -a_i \geqslant 2$ for $(p-i)=1$ is ${100 \choose 5} - {96 \choose 1}$. Please see that am I right or wrong. Also another approches are also welcome. Thank you in advance
