JEE 2023 — Mathematics PYQ

Step 1: Digits ko analyze karein

Diya gaya number hai: $123412341$

Ismein total 9 digits hain:

• Odd digits: $1,1,1,3,3$ (Total 5 digits)

• Even digits: $2,2,4,4$ (Total 4 digits)

Step 2: Places ko analyze karein

Ek 9-digit number mein positions aisi hoti hain:

$\text{O E O E O E O E O}$ (Jahan O = Odd place, E = Even place)

• Odd places: $1^{st},3^{rd},5^{th},7^{th},9^{th}$ (Total 5 places)

• Even places: $2^{nd},4^{th},6^{th},8^{th}$ (Total 4 places)

Step 3: Arrangement ki conditions

Sawal ke mutabik, even digits sirf even places par honi chahiye.

Humaare paas 4 even digits ($2,2,4,4$) hain aur exact 4 even places hain.

1. Even digits ko arrange karne ke tarike:

   Humaare paas 4 positions hain aur digits hain $2,2,4,4$.

   $$\text{Ways}=\frac{4!}{2!\times 2!}=\frac{24}{4}=6$$

2. Odd digits ko arrange karne ke tarike:

   Bachi hui 5 odd places par 5 odd digits ($1,1,1,3,3$) ko arrange karna hai.

   $$\text{Ways}=\frac{5!}{3!\times 2!}=\frac{120}{6\times 2}=10$$

Step 4: Total Numbers

Dono events independent hain, isliye total numbers honge:

Final Answer:

Aise total 60 numbers banaye ja sakte hain.