Verified Solution • 2025–26 Session moderate

QJEEAdv21Ma3471: JEE Advanced 2021 Mathematics - Class 11 JEEAdv21Ma3471 Solution: Let S 1 = { ( i , j , k ) : i , j , k ∈ { 1 , 2 , . . . . , 10…

Let

S1={(i,j,k):i,j,k∈{1,2,....,10}},

S2={(i,j):1≤i<j+2≤10,i,j∈{1,2,...,10}},

S3={(i,j,k,l):1≤i<j<k<l,i,j,k,l∈{1,2,...,10}} and

S4={(i,j,k,l):i,j,k and l are distinct elements in {1, 2, ...., 10}.

If the total number of elements in the set Sr is nr, r = 1, 2, 3, 4, then which of the following statements is(are) TRUE?

Multiple Choice Options

  • A n1 = 1000
  • B n2 = 44
  • C n3 = 220
  • D n412=420

Step-by-Step Verified Solution

n1 = number of elements in S1 = 10 × 10 × 10 = 1000

n2 = number of elements in S2 = 8 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1

= 8 + 8×92=44 (∵ 1 ≤ i < j + 2 ≤ 10 ⇒ j ≤ 8 and i ≥ 1

∴ (i = 1, j = 1, 2, 3, .... 8), (i = 2, j = 1, 2, 3, ..... 8), (i = 3, j = 2, 3, 4, .... 8) and so on)

n3 = number of elements in S3 = 10C4

(selecting 4 numbers and arranging in increasing order)

=10×9×8×724=210

n4 = number of elements in S4 = 10C4 = 10 × 9 × 8 × 7

⇒n412=420

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