Permutations and Combinations · Mathematics · JEE Advanced
Numerical
Let $S$ be the set of all seven-digit numbers that can be formed using the digits $0, 1$ and $2$. For example, $2210222$ is in $S$, but $0210222$ is NOT in $S$.
Then the number of elements $x$ in $S$ such that at least one of the digits $0$ and $1$ appears exactly twice in $x$, is equal to ____________.
A group of 9 students, $s_1, s_2, \ldots, s_9$, is to be divided to form three teams $X, Y$, and $Z$ of sizes 2,3 , and 4 , respectively. Suppose that $s_1$ cannot be selected for the team $X$, and $s_2$ cannot be selected for the team $Y$. Then the number of ways to form such teams, is ____________.
Then the number of such points for which $$x^2 + {y^2} + {z^2} \le 100$$ is
MCQ (Single Correct Answer)
(i) Let $$\alpha $$1 be the total number of ways in which the committee can be formed such that the committee has 5 members, having exactly 3 boys and 2 girls.
(ii) Let $$\alpha $$2 be the total number of ways in which the committee can be formed such that the committee has at least 2 members, and having an equal number of boys and girls.
i) Let $$\alpha $$3 be the total number of ways in which the committee can be formed such that the committee has 5 members, at least 2 of them being girls.
(iv) Let $$\alpha $$4 be the total number of ways in which the committee can be formed such that the committee has 4 members, having at least 2 girls such that both M1 and G1 are NOT in the committee together.
LIST-I | LIST-II |
---|---|
P. The value of $\alpha_1$ is | 1. 136 |
Q. The value of $\alpha_2$ is | 2. 189 |
R. The value of $\alpha_3$ is | 3. 192 |
S. The value of $\alpha_4$ is | 4. 200 |
5. 381 | |
6. 461 |
Which of the following is correct?
The value of $${{b_6}}$$ is
Consider all possible permutations of the letters of the word ENDEANOEL. Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.
Column I | Column II | ||
---|---|---|---|
(A) | The number of permutations containing the word ENDEA is | (P) | 5! |
(B) | The number of permutations in which the letter E occurs in the first and the last position is | (Q) | 2 $$\times$$ 5! |
(C) | The number of permutations in which none of the letters D, L, N occurs in the last five positions is | (R) | 7 $$\times$$ 5! |
(D) | The number of permutations in which the letters A, E, O occur only in odd positions is | (S) | 21 $$\times$$ 5! |

MCQ (More than One Correct Answer)
$${S_1} = \left\{ {(i,j,k):i,j,k \in \{ 1,2,....,10\} } \right\}$$,
$${S_2} = \left\{ {(i,j):1 \le i < j + 2 \le 10,i,j \in \{ 1,2,...,10\} } \right\}$$,
$${S_3} = \left\{ {(i,j,k,l):1 \le i < j < k < l,i,j,k,l \in \{ 1,2,...,10\} } \right\}$$ and
$${S_4} = \{ (i,j,k,l):i,j,k$$ and $$l$$ are distinct elements in {1, 2, ...., 10}.
If the total number of elements in the set Sr is nr, r = 1, 2, 3, 4, then which of the following statements is(are) TRUE?
Subjective
(a) The women are in majority?
(b) The men are in majority?