Assertion :(A): Let $P (n) = (n + 1) (n + 2) (n + 3) \dots (n + r)$ then the greatest integer which divides $P (n)$ is $r!$ . Reason: (R): Product of $r$ consecutive numbers is divisible by $r!$ .

Both (A) & (R) are individually true & (R) is correct explanation of (A).

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Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).

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(A) is true but (R) is false.

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(A) is false but (R) is true.

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Solution

The correct option is A Both (A) & (R) are individually true & (R) is correct explanation of (A).
$P (n) = (n + 1) (n + 2) . . . . . . (n + r),$
here number of factors in $P (n)$ are $r$ , so $P (n)$ is divisible by $r!$ .

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DIRECTIONS (Qs. 52 AND 53): The following item consist of two statements: one labelled as the Assertion a) and the other as Reason (R). You are to examine these two statements carefully and select the answers to these items using the codes given below:

a) Both A and R are individually true but R is the correct explanation of A

b) Both A and R are individually true but R is not the correct explanation of A

c) A is true but R is false

d) Both A and R are false

53. Assertion a): Saluva Narasimha put an end to the old dynasty and assumed the royal title.

Reason

(R): He wanted to save the kingdom from further degeneration and disintegration. [2003]

a) Both A and R are individually true but R is the correct explanation of A

b) Both A and R are individually true but R is not the correct explanation of A

c) A is true but R is false

d) Both A and R are false

52. Assertion a): Shah Alam II spent the initial years as an emperor far away from his capital.

Reason

(R): There was always a lurking danger of foreign invasion from the north-west frontier. [2003]

Q. Assertion(A): Some algae are unicellular.
Reason(R): They belong to the kingdom Monera.

Q. Assertion (A): pH of pure water increases with increase in temperature. Reason (R): Self ionization of water is an endothermic reaction. Which of the following is correct?

Q. Assertion (A)

\sqrt{8}

is an irrational number.
Reason (R)
If m is a natural number which is not a perfect square, then

\sqrt{m}

is irrational.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

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Assertion :(A): Let P(n)=(n+1)(n+2)(n+3)…(n+r) then the greatest integer which divides P(n) is r!. Reason: (R): Product of r consecutive numbers is divisible by r!.

The correct option is A Both (A) & (R) are individually true & (R) is correct explanation of (A).P(n)=(n+1)(n+2)......(n+r), here number of factors in P(n) are r, so P(n) is divisible by r!.

Assertion :(A): Let $P (n) = (n + 1) (n + 2) (n + 3) \dots (n + r)$ then the greatest integer which divides $P (n)$ is $r!$ . Reason: (R): Product of $r$ consecutive numbers is divisible by $r!$ .

The correct option is A Both (A) & (R) are individually true & (R) is correct explanation of (A).
$P (n) = (n + 1) (n + 2) . . . . . . (n + r),$
here number of factors in $P (n)$ are $r$ , so $P (n)$ is divisible by $r!$ .