Line L of slope 2 and line L of slope 1 2 1 2 intersect at the origin O. In the first quadrant, P, P,$\ldots. P 1 2 12 are 12 points o_n line L and Q, Q,$$\ldots.. Q are 9 points o_n line L. Then the total number of triangles, that can be 1 1 2 9 2 formed having vertices at three of the 22 points O, P, P,$$\ldots P, Q, Q,$$\ldots. Q, is$: 1 2 12 1 2 9
Let $A=\{1,2,3,\ldots,10\}$, $B=\{4,8,12,16,20\}$ and $C=\{D:D\subseteq A,\,D\cap B\neq\emptyset\}$. The number of elements in $C$ which have at least 3 but at most 6 cardinal number is
Four-digit numbers are formed using digits from $\{0,1,2,3,4,5\}$, repetition allowed. Statement $S_1$: the number of such odd numbers is 480. Statement $S_2$: the number formed with exactly three different digits is 360.
If $p$, $q$, $r$ are prime numbers and $\alpha$, $\beta$, $\gamma$ are positive integers such that $\mathrm{lcm}(\alpha,\beta,\gamma)=p^3q^2r$ and $\gcd(\alpha,\beta,\gamma)=pqr$, then the number of possible ordered triplets $(\alpha,\beta,\gamma)$ is:
If $p$, $q$, $r$ are prime numbers and $\alpha$, $\beta$, $\gamma$ are positive integers such that $\mathrm{lcm}(\alpha,\beta,\gamma)=p^3q^2r$ and $\gcd(\alpha,\beta,\gamma)=pqr$, then the number of possible ordered triplets $(\alpha,\beta,\gamma)$ is:
Let $A=\{1,2,\ldots,26\}$. Match each type of relation on $A$ with the count formula and find $p+q$, $p+q+r$, etc.
**List-I:** P) Asymmetric relations=$p^q$ ($p$ prime), find $p+q$; Q) Reflexive but not symmetric $=p^q(p^q-p^r)$, find $p+q+r$; R) Symmetric but not reflexive (non-empty) $=p^q(p^r-1)-1$, find $p+q+r$; S) Neither symmetric nor reflexive (non-empty) $=(p^s-1)p^q(p^q-1)$, find $p+q+s$
**List-II:** 1)328; 2)329; 3)330; 4)353; 5)356
Let $A=\{1,2,\ldots,26\}$. Match each type of relation on $A$ with the count formula and find $p+q$, $p+q+r$, etc.
**List-I:** P) Asymmetric relations=$p^q$ ($p$ prime), find $p+q$; Q) Reflexive but not symmetric $=p^q(p^q-p^r)$, find $p+q+r$; R) Symmetric but not reflexive (non-empty) $=p^q(p^r-1)-1$, find $p+q+r$; S) Neither symmetric nor reflexive (non-empty) $=(p^s-1)p^q(p^q-1)$, find $p+q+s$
**List-II:** 1)328; 2)329; 3)330; 4)353; 5)356
Triplet \((x, y, z)\) is chosen from the set \(\{1,2,3, \ldots, n\}\), such that \(x \leq y \lt z\). The number of such triplets is
The number of ways of selecting two numbers a and b, \(a \in \{2, 4, 6, \ldots, 100\}\) and \(b \in \{1, 3, 5, \ldots, 99\}\) such that 2 is the remainder when a + b is divided by 23 is