JEE 2023 — Mathematics PYQ

Step 1: Courses ko divide karein

• Language Courses: $5$

• Other Courses (Non-language): $12-5=7$

• Total select karne hain: $5$

Step 2: Condition apply karein ("At most two language courses")

"At most 2" ka matlab hai ki language courses $0,1,$ ya $2$ ho sakte hain. Iske liye hum teen cases banayenge:

1. Case 1: 0 Language aur 5 Others

   $$\text{Ways}=\left(\genfrac{}{}{0ex}{}{5}{0}\right)\times \left(\genfrac{}{}{0ex}{}{7}{5}\right)=1\times 21=21$$

2. Case 2: 1 Language aur 4 Others

   $$\text{Ways}=\left(\genfrac{}{}{0ex}{}{5}{1}\right)\times \left(\genfrac{}{}{0ex}{}{7}{4}\right)=5\times 35=175$$

3. Case 3: 2 Languages aur 3 Others

   $$\text{Ways}=\left(\genfrac{}{}{0ex}{}{5}{2}\right)\times \left(\genfrac{}{}{0ex}{}{7}{3}\right)=10\times 35=350$$

Step 4: Total ways nikalna

Saare cases ko jodne par:

Final Answer:

Woh boy 546 tariko se courses choose kar sakta hai.