JEE Main · 2022 · 27 Jun 2022 · Shift 2 · Permutations and Combinations

The four entries $a,b,c,d$ each range over $\{0,1,2,3,4,5\}$ and their sum must be a prime strictly between $2$ and $8$, that is $3$, $5$ or $7$. Counting non-negative integer solutions by stars and bars, a sum of $3$ gives $\left(\genfrac{}{}{0ex}{}{6}{3}\right)=20$ and a sum of $5$ gives $\left(\genfrac{}{}{0ex}{}{8}{3}\right)=56$, and in both the totals are small enough that no entry can exceed $5$. For a sum of $7$ the unrestricted count is $\left(\genfrac{}{}{0ex}{}{10}{3}\right)=120$, from which the cases with an entry of at least $6$ must be removed: fixing one entry to be $\ge 6$ leaves the remaining three to sum to $1$ in $\left(\genfrac{}{}{0ex}{}{4}{3}\right)=4$ ways, and with four choices of which entry that is, this removes $4\times 4=16$ (two entries exceeding $5$ is impossible when the total is only $7$), leaving $120-16=104$. The total is $20+56+104=180$.