🎓 JEE MAIN📅 Year: 2023📚 Mathematics🏷 Permutations and Combinations
3
The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is :
A bag contains 6 white and 4 black balls. A die is rolled once and the number of balls equal to the number obtained on the die are drawn from the bag at random. The probability that all the balls drawn are white is:
Consider a function $f:\mathbb{N}\to\mathbb{R}$ satisfying
\[
f(1)+2f(2)+3f(3)+\cdots+xf(x)=x(x+1)f(x),\quad x\ge 2,
\]
with $f(1)=1$. Then
\[
\frac{1}{f(2022)}+\frac{1}{f(2028)}
\]
is equal to:
Let \( S=\{w_1,w_2,\ldots\} \) be the sample space of a random experiment.
Let the probabilities satisfy
\[
P(w_n)=\frac{P(w_{n-1})}{2},\qquad n\ge 2.
\]
Let
\[
A=\{\,2k+3\ell : k,\ell\in\mathbb{N}\,\},\qquad
B=\{\,w_n : n\in A\,\}.
\]
Then \(P(B)\) is equal to:
Let $A_1$ and $A_2$ be two arithmetic means and $G_1, G_2, G_3$ be three geometric means of two distinct positive numbers. Then $G_1^4+G_2^4+G_3^4+G_1^2G_3^2$ is equal to: