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Permutations and Combinations · Question 31

JEE Main 2025 (Online) 29th January Morning ShiftMCQEasy📈 Topic in 23% papers+4−1
1

Let P be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in P are formed by using the digits 1, 2 and 3 only, then the number of elements in the set P is :

  1. 164
  2. 158
  3. 161
  4. 173

Answer

(3) 161

Solution

Let the number contain a ones, b twos and c threes.

a + b + c = 7 and a + 2b + 3c = 11

b + 2c = 4

Possible cases:

(i) c = 0, b = 4, a = 3: 7!3! 4! = 35

(ii) c = 1, b = 2, a = 4: 7!4! 2! 1! = 105

(iii) c = 2, b = 0, a = 5: 7!5! 2! = 21

n(P) = 35 + 105 + 21 = 161

Answer: (3)

JEE Main 2025 (Online) 29th January Morning Shift

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2

The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is .

JEE Main 2025 (Online) 29th January Morning Shift
3

Let nCr-1 = 28, nCr = 56 and nCr+1 = 70. Let A(4 t, 4 t), B(2 t, -2 t) and C (3r - n, r2 - n - 1 ) be the vertices of a triangle ABC, where t is a parameter. If (3x - 1)2 + (3y)2 = , is the locus of the centroid of triangle ABC, then equals

  1. 18
  2. 8
  3. 20
  4. 6
JEE Main 2025 (Online) 28th January Morning Shift
4

The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is

  1. 5720
  2. 5719
  3. 4608
  4. 4607
JEE Main 2025 (Online) 28th January Morning Shift
5

The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is .

JEE Main 2025 (Online) 28th January Evening Shift

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