If the total number of elements in the set S r is n r , r = 1 , 2 , 3 , 4 , then which of the following statements is (are) TRUE? [2021]

Question

Let $S_{1} = {(i, j, k) : i, j, k \in {1, 2, \dots, 10}}$

$S_{2} = {(i, j) : 1 \leq i < j + 2 \leq 10, i, j \in {1, 2, \dots, 10}}$

$S_{3} = {(i, j, k, l) : 1 \leq i < j < k < l, i, j, k, l \in {1, 2, \dots, 10}}$

and $S_{4} = {(i, j, k, l) : i, j, k and l are distinct elements in {1, 2, \dots, 10}}$ .

If the total number of elements in the set $S_{r}$ is $n_{r}$ , $r = 1, 2, 3, 4$ , then which of the following statements is (are) TRUE? [2021]

Smart Achievers · Accepted Answer

(1, 2, 4)

Number of elements in $S_{1} = 10 \times 10 \times 10 = 1000$

Number of elements in $S_{2} = 9 (J = 8) + 8 (J = 7) + 7 (J = 6) + 6 (J = 5) + 5 (J = 4) + 4 (J = 3) + 3 (J = 2) + 2 (J = 1) = 44$

Number of elements in $S_{3} = {}^{10}C_{4} = 210$

Number of elements in $S_{4} = {}^{10}P_{4} = 210 \times 4! = 5040$

So, options (1), (2), (4) are correct.

Solution

1 $n_{1} = 1000$

2 $n_{2} = 44$

3 $n_{3} = 220$

4 $\frac{n_{4}}{12} = 420$

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Solution

1 n1=1000

2 n2=44

3 n3=220

4 n412=420

Ans. (1, 2, 4) Number of elements in S1=10×10×10=1000 Number of elements in S2=9(J=8)+8(J=7)+7(J=6)+6(J=5)+5(J=4)+4(J=3)+3(J=2)+2(J=1)=44 Number of elements in S3=C410=210 Number of elements in S4=P410=210×4!=5040 So, options (1), (2), (4) are correct.

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1 $n_{1} = 1000$

2 $n_{2} = 44$

3 $n_{3} = 220$

4 $\frac{n_{4}}{12} = 420$