JEE Advanced · Mathematics · 4. P&C

Consider $4$ boxes, where each box contains $3$ red balls and $2$ blue balls. Assume that all $20$ balls are distinct. In how many different ways can $10$ balls be chosen from these $4$ boxes so that from each box at least one red ball and one blue ball are chosen?

A $21816$
B $85536$
C $12096$
D $156816$

Verified Solution

Answer & Solution

Correct Answer

(A) $21816$

Step-by-step Solution

Detailed explanation

Let the four boxes be Box-1, Box-2, Box-3 and Box-4. Each of the box has

3

red balls and

2

blue balls.

10

balls can be chosen in:
Case I - When exactly one box gives four balls
Case II - When exactly two boxes gives three balls
Required number of ways

=

Number of ways in Case I

+

Number of ways in Case II.
$={ }^4 C_1\left[{ }^2 C_1+{ }^3 C_1\right]\left({ }^3 C_1{ }^2 C_1\right)^3+{ }^4 C_2[{ }^3 C_2{ }^2 C_1$ $+{ }^3 C_1{ }^2 C_2]^2\left[{ }^3 C_1{ }^2 C_1\right]^2$

= 4 (5) {(6)}^{3} + 6 {(3 \times 2 + 3)}^{2} {(6)}^{2}

= 4320 + 17496 = 21816

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.

Column I	Column II
(A) The minimum value of $\frac{x^2+2 x+4}{x+2}$ is	(p) 0
(B) Let $A$ and $B$ be $3 \times 3$ matrices of real numbers, where $A$ is symmetric, $B$ is skew-symmetric and $(A+B)(A-B) =$ $(A-B)(A+B)$. If $(A B)^t=(-1)^k$ $A B$, where $(A B)^t$ is the transpose of the matrix $A B$, then possible values of $k$ are	(q) 1
(C) Let $a=\log _3 \log _3 2$. An integer $k$ satisfying $1<2^{\left(-k+3^{-a}\right)}<2$, must be less than	(r) 2
(D) If $\sin \theta=\cos \phi$, then the possible values of $\frac{1}{\pi}\left(\theta \pm \phi-\frac{\pi}{2}\right)$ are	(s) 3

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If $f(x)=\int_{0}^{x} e^{t^{2}}(t-2)(t-3) d t$ for all $x \in(0, \infty)$, then

JEE Advanced 2012 Hard

Consider the given data with frequency distribution

\begin{array}{l} x_{i} & 3 & 8 & 11 & 10 & 5 & 4 \\ f_{i} & 5 & 2 & 3 & 2 & 4 & 4 \end{array}

Match each entry in List-I to the correct entries in List-II.

	List-I		List-II
$(P)$	The mean of the above data is	$(1)$	$2.5$
$(Q)$	The median of the above data is	$(2)$	$5$
$(R)$	The mean deviation about the mean of the above data is	$(3)$	$6$
$(S)$	The mean deviation about the median of the above data is	$(4)$	$2.7$
		$(5)$	$2.4$

The correct option is

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Column I	Column II
(A) The minimum value of \(\frac{x^2+2 x+4}{x+2}\) is	(p) 0
(B) Let \(A\) and \(B\) be \(3 \times 3\) matrices of real numbers, where \(A\) is symmetric, \(B\) is skew-symmetric and \((A+B)(A-B) =\) \((A-B)(A+B)\). If \((A B)^t=(-1)^k\) \(A B\), where \((A B)^t\) is the transpose of the matrix \(A B\), then possible values of \(k\) are	(q) 1
(C) Let \(a=\log _3 \log _3 2\). An integer \(k\) satisfying \(1<2^{\left(-k+3^{-a}\right)}<2\), must be less than	(r) 2
(D) If \(\sin \theta=\cos \phi\), then the possible values of \(\frac{1}{\pi}\left(\theta \pm \phi-\frac{\pi}{2}\right)\) are	(s) 3

Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from these 4 boxes so that from each box at least one red ball and one blue ball are chosen?

Step-by-step Solution

Consider $4$ boxes, where each box contains $3$ red balls and $2$ blue balls. Assume that all $20$ balls are distinct. In how many different ways can $10$ balls be chosen from these $4$ boxes so that from each box at least one red ball and one blue ball are chosen?