JEE Mathematics 2025 Question with Solution

Given: A pack has $52$ cards with $13$ spades. One card is lost. From the remaining cards, $n$ cards are drawn and all are found to be spades.

Find: The value of $n$ if the probability that the lost card is a spade is $\frac{11}{50}$.

If $n$ drawn cards are all spades, then the remaining number of spades is $13-n$ and the remaining total number of cards is $52-n$.

Therefore,

$$P(\text{lost card is spade})=\frac{\binom{13-n}{1}}{\binom{52-n}{1}}=\frac{11}{50}$$

Since $\binom{a}{1}=a$, this becomes

$$\frac{13-n}{52-n}=\frac{11}{50}$$

Cross-multiplying,

$$50(13-n)=11(52-n)$$ $$650-50n=572-11n$$ $$78=39n$$ $$n=2$$

Therefore, the value of $n$ is $2$. The correct option is B.

Given: After $n$ cards are drawn and all are spades, the unseen cards contain $13-n$ spades out of $52-n$ total cards.

Find: $n$.

The probability that the lost card is a spade is the fraction of spades among the unseen cards:

$$\frac{13-n}{52-n}=\frac{11}{50}$$

Now solve step by step:

$$50(13-n)=11(52-n)$$ $$650-50n=572-11n$$ $$650-572=50n-11n$$ $$78=39n$$ $$n=2$$

Hence, $n=2$.