1
MCQ (Single Correct Answer)

JEE Main 2021 (Online) 1st September Evening Shift

Let P1, P2, ......, P15 be 15 points on a circle. The number of distinct triangles formed by points Pi, Pj, Pk such that i +j + k $$\ne$$ 15, is :
A
12
B
419
C
443
D
455

Explanation

Total number of triangles = $${}^{15}{C_3}$$

i + j + k = 15 (Given)


Number of possible triangles using the vertices Pi, Pj, Pk such that i + j + k $$\ne$$ 15 is equal to $${}^{15}{C_3}$$ $$-$$ 12 = 443

Option (c)
2
MCQ (Single Correct Answer)

JEE Main 2021 (Online) 25th July Evening Shift

If $${}^n{P_r} = {}^n{P_{r + 1}}$$ and $${}^n{C_r} = {}^n{C_{r - 1}}$$, then the value of r is equal to :
A
1
B
4
C
2
D
3

Explanation

$${}^n{P_r} = {}^n{P_{r + 1}} \Rightarrow {{n!} \over {(n - r)!}} = {{n!} \over {(n - r - 1)!}}$$

$$ \Rightarrow (n - r) = 1$$ .....(1)

$${}^n{C_r} = {}^n{C_{r - 1}}$$

$$ \Rightarrow {{n!} \over {r!(n - r)!}} = {{n!} \over {(r - 1)!(n - r + 1)!}}$$

$$ \Rightarrow {1 \over {r(n - r)!}} = {1 \over {(n - r + 1)(n - r)!}}$$

$$ \Rightarrow n - r + 1 = r$$

$$ \Rightarrow n + 1 = 2r$$ ..... (2)

From (1) and (2), $$ 2r - 1 - r = 1 \Rightarrow r = 2$$
3
MCQ (Single Correct Answer)

JEE Main 2021 (Online) 18th March Morning Shift

The sum of all the 4-digit distinct numbers that can be formed with the digits 1, 2, 2 and 3 is :
A
26664
B
122664
C
122234
D
22264

Explanation

Total possible numbers using 1, 2, 2 and 3 is

= $${{4!} \over {2!}}$$ = 12

When unit place is 1, the total possible numbers using remaining 2, 2 and 3 are

= $${{3!} \over {2!}}$$ = 3

When unit place is 2, the total possible numbers using remaining 1, 2 and 3 are

= 3! = 6

When unit place is 3, the total possible numbers using remaining 1, 2 and 2 are

= $${{3!} \over {2!}}$$ = 3

$$ \therefore $$ Sum of unit places of all (3 + 6 + 3) 12 numbers is

= ( 1$$ \times $$3 + 2$$ \times $$6 + 3$$ \times $$3)

Similarly,

When 10th place is 1, the total possible numbers using remaining 2, 2 and 3 are

= $${{3!} \over {2!}}$$ = 3

When 10th place is 2, the total possible numbers using remaining 1, 2 and 3 are

= 3! = 6

When 10th place is 3, the total possible numbers using remaining 1, 2 and 2 are

= $${{3!} \over {2!}}$$ = 3

$$ \therefore $$ Sum of 10th places of all (3 + 6 + 3) 12 numbers is

= ( 1$$ \times $$3 + 2$$ \times $$6 + 3$$ \times $$3) $$ \times $$ 10

Similarly,

Sum of 100th places of all (3 + 6 + 3) 12 numbers is

= ( 1$$ \times $$3 + 2$$ \times $$6 + 3$$ \times $$3) $$ \times $$ 100

and Sum of 1000th places of all (3 + 6 + 3) 12 numbers is

= ( 1$$ \times $$3 + 2$$ \times $$6 + 3$$ \times $$3) $$ \times $$ 1000

$$ \therefore $$ Total sum = ( 1$$ \times $$3 + 2$$ \times $$6 + 3$$ \times $$3) + ( 1$$ \times $$3 + 2$$ \times $$6 + 3$$ \times $$3) $$ \times $$ 10

+ ( 1$$ \times $$3 + 2$$ \times $$6 + 3$$ \times $$3) $$ \times $$ 100 + ( 1$$ \times $$3 + 2$$ \times $$6 + 3$$ \times $$3) $$ \times $$ 1000

= (3 + 12 + 9) (1 + 10 + 100 + 1000) = 1111 $$ \times $$ 24 = 26664
4
MCQ (Single Correct Answer)

JEE Main 2021 (Online) 17th March Evening Shift

If the sides AB, BC and CA of a triangle ABC have 3, 5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to :
A
240
B
360
C
333
D
364

Explanation


Total number of triangles

= $${}^{14}{C_3} - {}^3{C_3} - {}^5{C_3} - {}^6{C_3}$$

= 364 – 31 = 333

Questions Asked from Permutations and Combinations

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