'PERMUTATIONS' be arranged if the

(i) words start with P and ends with S and 

(ii) Vowels are all together.

Solution:

Explanation: In the word PERMUTATIONS, there are two T's  and all the other letters appear only once.

(i) If p and s are fixed at the extreme ends, then 10 letters are left. 

p_ _ _ _ _ _ s

10!/2! =10×9×8×7×6×5×4×3×2!  / 2!

= 1814400

So , in this case, the required number of arrangements

(ii) There are five vowels in the given word, each appearing only once.

Since they have to always occur together, they are treated as a single object.

So, a single object together with the remaining 7 objects will be 8 objects. 

These 8 objects in which there are two  T's  can be arranged in 8! / 2!

Corresponding to each of these arrangements, the 5 different vowels can be arranged in 5! ways. 

Therefore, the required number of arrangements 

8! /2!× 5! =8×7×6×5×4×3×2! / 2! × 120

= 2419200

Final Answer: (i) The required number of arrangements when  T' and S' are fixed at the extreme ends is 1814400.

(ii) The required number of arrangements when all vowels are together is 2419200.

Solution:

Explanation:

A line is a one-dimensional object of infinite length with no width or thickness that

never bends or turns and defined as a line of points that extends infinitely in two directions.

A point is an object without shape or size, used to describe a location or position.

So, from one point there is an uncountable number of lines that can pass through.

Hence, an infinite number of lines can pass through one point.

Final Answer:

An infinite number of lines can pass through one point.

A: Ichthyology

B: Herpetology

C: Serpentology

D: Ornithology

Solution:

Explanation

Herpetology, a branch of zoology that deals with the study of amphibians and reptiles,

provide significant insight into the terrestrial and aquatic ecosystem.

Ichthyology: The branch of zoology that deals with fishes.

Serpentology: A branch of zoology that studies snakes.

Ornithology: A branch of zoology dealing with the study of birds.

Final Answer

The correct option is option B) Herpetology.

 If the statement is false rewrite the given statement correctly

(i) If P = {m, n} and Q = {n, m} then  P × Q = {(m, n), (n, m)}

(ii) If A and B are non - empty sets then  A × B is a non - empty set of ordered pairs (x, y) such that  x∈ A and  y∈ B

(iii) If A = {1, 2}, B = {3, 4} then  A ×  (B∩ϕ) = ϕ

Solution:

Explanation:

for r=1,2,3 .If for some positive integers m, n we have 

Solution:

Explanation:

Let, common difference =d

Final Answer:

Option (C) is correct which is 1.