Question:
There are 10 cards numbered \(1\) to \(10\). The number of ways in which at least 3 cards can be chosen from these 10 cards is
There are 10 cards numbered \(1\) to \(10\). The number of ways in which at least 3 cards can be chosen from these 10 cards is
Show Hint
For "at least" type counting problems, it is usually easier to count the total number of possibilities and subtract the unwanted cases.
Updated On: Jul 29, 2026
- \(1023\)
- \(1013\)
- \(1024\)
- \(968\)
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Concept:
The total number of selections from \(n\) distinct objects is \(2^n\). To find the number of selections containing at least 3 objects, subtract the selections containing fewer than 3 objects.
Step 1: Find the total number of ways to choose cards. Since there are \(10\) distinct cards, \[ \text{Total selections} = 2^{10} = 1024. \]
Step 2: Find the number of selections containing fewer than 3 cards. Choosing \(0\) cards: \[ {10 \choose 0}=1. \] Choosing \(1\) card: \[ {10 \choose 1}=10. \] Choosing \(2\) cards: \[ {10 \choose 2}=45. \] Hence, \[ 1+10+45=56. \]
Step 3: Find the number of selections containing at least 3 cards. \[ 1024-56=968. \] Therefore, the required number of ways is \[ \boxed{968} \] \[ \boxed{\text{Answer = (D)}} \]
Step 1: Find the total number of ways to choose cards. Since there are \(10\) distinct cards, \[ \text{Total selections} = 2^{10} = 1024. \]
Step 2: Find the number of selections containing fewer than 3 cards. Choosing \(0\) cards: \[ {10 \choose 0}=1. \] Choosing \(1\) card: \[ {10 \choose 1}=10. \] Choosing \(2\) cards: \[ {10 \choose 2}=45. \] Hence, \[ 1+10+45=56. \]
Step 3: Find the number of selections containing at least 3 cards. \[ 1024-56=968. \] Therefore, the required number of ways is \[ \boxed{968} \] \[ \boxed{\text{Answer = (D)}} \]
Was this answer helpful?
0
0
Top AP EAPCET Mathematics Questions
- $D = \left\{x\in\mathbb{R}: f\left(x\right) =\sqrt{\frac{x - \left|x\right|}{x - \left[x\right]}} \text{is defined} \right\}$ and $C$ be the range of the real function $g(x) = \frac{2x}{4 + x^{2}}$. Then $D \cap C$
- $f\left(x\right) = \frac{x}{e^{x}-1} + \frac{x}{2} + 2 \cos^{3} \frac{x}{2} $ on $R-\left\{0\right\} $ is
- Consider the following lists.
List-I List-II (A) $f(x) = \frac{|x+2|}{x+2} , x \ne -2 $ (I) $[\frac{1}{3} , 1 ]$ (B) $(x)=|[x]|,x \in [R$ (II) Z (C) $h(x) = |x - [x]| , x \in [R$ (III) W (D) $f(x) = \frac{1}{2 - \sin 3x} , x \in [R$ (IV) [0, 1) (V) { -1, 1} - If $[x]$ is the greatest integer less than or equal to $x$ and $|x|$ is the modulus of $x$. then the system of three equations $2x + 3 | y | + 5[z] = 0, x + |y| - 2[z] = 4, x + |y| + |z| = 1$ has
- Investigate the values of $\lambda$ and $\mu$ for the system $x+2 y+3 z=6, x+3 y+5 z=9$, $2 x+5 y+\lambda z=\mu$ and match the values in List - I with the items in List - II.
List I List II (A) $\lambda=8, \mu \neq 15$ 1. Infinitely many solutions (B) $\lambda \neq 8, \mu \in R$ 2. No solution (C) $\lambda=8, \mu=15$ 3. Unique solution
View More Questions
Top AP EAPCET Combinations Questions
- The number of six-digit natural numbers that can be formed with the digits 2, 3, 4, 0, 5, 6, 7, 8 is:
- The number of ways in which a committee of 7 members can be formed from 6 teachers, 5 fathers and 4 students in such a way that at least one from each group is included and teachers form the majority among them, is:
- If $^{n}C_{12} = ^{n}C_{8}$, then $^{n}C_{17} =$
- A student is to answer \(10\) out of \(13\) questions in an examination such that he must choose at least \(4\) from the first \(5\) questions. The number of choices available to him is:
- The least value of \(n\) so that \[ {}^{\,n-1}C_3+{}^{\,n-1}C_4\gt {}^{\,n}C_3 \] is
View More Questions
Top AP EAPCET Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- The four quantum numbers of the valence electron of potassium are :
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The volume of $0.02\, M$ acidified permanganate solution required for complete reaction of $60\, mL$ of $0.01\, MI^{-}$ ion solution to form $I_2$ in $mL$ is
View More Questions