First slide

Combinations

Question

$Let A = \{\frac{1}{10}, \frac{1}{9}, \frac{1}{8}, \dots \dots \dots, \frac{1}{3}, \frac{1}{2}, 1, 2, 3, 4, \dots \dots .8, 9, 10\}, then the number of ordered pairs (a, b) such that a, b \in A and$

Difficult

A

$\log_{b}^{} a \leq 0 is 180$
B

$\log_{b}^{} a > 0 is 162$
C

$\log_{b}^{} a < 0 is 162$
D

$All the above$

Solution

$\begin{array}{l} (\log_{b} a) < 0 if a > 1, 0 < b < 1 (or) 0 < a < 1, b > 1 \\ Number of pairs (a, b) = (^{9} C_{1} \times^{9} C_{1}) \times 2 = 162 \\ (\log_{b} a) = 0 if a = 1, b \neq 1, b > 0 \end{array}$
$\begin{array}{l} Number of pairs (a, b) = 18 \\ (\log_{b} a) \leq 0 \Rightarrow number of ordered pairs (a, b) = 180 \\ (\log_{b} a) > 0 if a > 1, b > 1 (or) 0 < a < 1, 0 < b < 1 \end{array}$
$Number of ordered pairs = (9 \times 9) \times 2 = 162$
Therefore, the correct answer is (D).

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