(C) Total number of ways to draw $6$ cards from $52$ cards is $^{52}C_6$.Number of red cards in a pack is $26$ and number of black cards is $26$.We ne

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$\frac{^{26}C_3 \times ^{26}C_3}{^{52}C_6}$

(C) Total number of ways to draw $6$ cards from $52$ cards is $^{52}C_6$.Number of red cards in a pack is $26$ and number of black cards is $26$.We need to choose $3$ red cards from $26$ and $3$ black cards from $26$.Number of ways to choose $3$ red cards is $^{26}C_3$.Number of ways to choose $3$ black cards is $^{26}C_3$.Therefore,the required probability is $\frac{^{26}C_3 \times ^{26}C_3}{^{52}C_6}$.

Class 11 Mathematics Probability Advanced Use of permutations and combinations in probability

Easy

Six cards are drawn simultaneously from a pack of playing cards. What is the probability that $3$ will be red and $3$ will be black?

A
$^{26}C_6$
B
$\frac{^{26}C_3}{^{52}C_6}$
C
$\frac{^{26}C_3 \times ^{26}C_3}{^{52}C_6}$
D
$\frac{1}{2}$

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Probability

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Advanced Use of permutations and combinations in probability

From the first $100$ natural numbers, two numbers $a$ and $b$ are selected randomly without replacement. If the probability that $a-b \ge 10$ is $\frac{m}{n}$, with $\gcd(m, n)=1$, then $m+n$ is equal to:

DifficultJEE MAIN 2026

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In a shoe rack,there are $4$ pairs of shoes. If $4$ shoes are drawn one after the other at random without replacement,what is the probability of getting at least one correct pair of shoes among the four shoes drawn?

MediumAP EAMCET 2025

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$A$ box contains $12$ balls,of which $4$ are red,$5$ are green,and $3$ are white. If $3$ balls are drawn at random simultaneously from the box,then the probability that exactly $2$ balls have the same colour is

MediumAP EAMCET 2025

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From a group of $7$ men and $4$ ladies,a committee of $6$ persons is formed. What is the probability that the committee contains exactly $2$ ladies?

Easy

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$A$ drawer contains $5$ brown socks and $4$ blue socks well mixed. $A$ man reaches the drawer and pulls out $2$ socks at random. What is the probability that they match?

Medium

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Six cards are drawn simultaneously from a pack of playing cards. What is the probability that $3$ will be red and $3$ will be black?

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From the first $100$ natural numbers, two numbers $a$ and $b$ are selected randomly without replacement. If the probability that $a-b \ge 10$ is $\frac{m}{n}$, with $\gcd(m, n)=1$, then $m+n$ is equal to:

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