JEE MAIN - Mathematics (2020 - 3rd September Evening Slot - No. 10)
The probability that a randomly chosen 5-digit
number is made from exactly two digits is :
$${{150} \over {{{10}^4}}}$$
$${{134} \over {{{10}^4}}}$$
$${{121} \over {{{10}^4}}}$$
$${{135} \over {{{10}^4}}}$$
Explanation
Sample space = 9 $$ \times $$ 104
Case - I
Out of exactly two digits selected one is zero then favourable cases = $${}^9{C_1}({2^4} - 1)$$
Case - II
Both selected digits are non-zero then favourable cases = $${}^9{C_2}({2^5} - 2)$$
Probability = $${{9({2^4} - 1) + {{9.8} \over 2}({2^5} - 2)} \over {9 \times {{10}^4}}}$$
$$ = {{15 + 120} \over {{{10}^4}}} = {{135} \over {{{10}^4}}}$$
Case - I
Out of exactly two digits selected one is zero then favourable cases = $${}^9{C_1}({2^4} - 1)$$
Case - II
Both selected digits are non-zero then favourable cases = $${}^9{C_2}({2^5} - 2)$$
Probability = $${{9({2^4} - 1) + {{9.8} \over 2}({2^5} - 2)} \over {9 \times {{10}^4}}}$$
$$ = {{15 + 120} \over {{{10}^4}}} = {{135} \over {{{10}^4}}}$$
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