Let \( S \) be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set \( S \), one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is:
Show Hint
- \( \frac{1}{4} \)
- \( \frac{2}{3} \)
- \( \frac{1}{3} \)
- \( \frac{1}{2} \)
The Correct Option is D
Solution and Explanation
We are given the word GARDEN, which consists of the following letters: G, A, R, D, E, N. Among these letters, the vowels are A and E. To find the probability that the selected word will NOT have vowels in alphabetical order, we proceed as follows:
Step 1: Total number of arrangements.
Since there are 6 distinct letters in the word GARDEN, the total number of ways to arrange these letters is: \[ {Total arrangements} = 6! = 720 \]
Step 2: Number of favorable cases (vowels in alphabetical order).
For the vowels A and E to be in alphabetical order, the positions of A and E must be such that A appears before E. The total number of ways to arrange the 6 letters such that A appears before E is: \[ {Favorable cases} = \binom{6}{2} \cdot 4! = 15 \cdot 24 = 360 \]
Step 3: Probability calculation.
The probability that the selected word will have vowels in alphabetical order is: \[ P = \frac{360}{720} = \frac{1}{2} \] Therefore, the probability that the selected word will NOT have vowels in alphabetical order is: \[ P({Not in order}) = 1 - \frac{1}{2} = \frac{1}{2} \]
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
Top JEE Main Set Theory Questions
- A group of 40 students appeared in an examination of 3 subjects – Mathematics, Physics & Chemistry. It was found that all students passed in at least one of the subjects, 20 students passed in Mathematics, 25 students passed in Physics, 16 students passed in Chemistry, at most 11 students passed in both Mathematics and Physics, at most 15 students passed in both Physics and Chemistry, at most 15 students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is ________.
- Let \( A = \{1, 2, 3, \ldots, 7\} \) and let \( P(A) \) denote the power set of \( A \). If the number of functions \( f : A \rightarrow P(A) \) such that \( a \in f(a), \, \forall a \in A \) is \( m^n \), \( m \) and \( n \in \mathbb{N} \) and \( m \) is least, then \( m + n \) is equal to \(\_\_\_\_\_\_\_\_\_\).
- Let \( A \) and \( B \) be two finite sets with \( m \) and \( n \) elements respectively. The total number of subsets of the set \( A \) is 56 more than the total number of subsets of \( B \). Then the distance of the point \( P(m, n) \) from the point \( Q(-2, -3) \) is:
- Let the set \( C = \{(x, y) \mid x^2 - 2^y = 2023, x, y \in \mathbb{N}\} \).Then\[\sum_{(x, y) \in C} (x + y)\]is equal to ______.
- The number of symmetric relations defined on the set {1, 2, 3, 4} which are not reflexive is _____.
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.