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14.Probability
hard
Consider the set of all $7-$digit numbers formed by the digits $0,1,2,3,4,5,6$, each chosen exactly once. If a number is randomly drawn from this set, the probability that it is divisible by $4$ is
A
$\frac{26}{105}$
B
$\frac{13}{45}$
C
$\frac{2}{7}$
D
$\frac{1}{3}$
(KVPY-2021)
Solution
(b)
$\{0,1,2,3,4,5,6\}$
Total $=7 !-6 !=6.6 !$
fav : last two digit $\{12,16,24,32,36,52,56,64\}$
$8(5 !-4 !)$
last two digit $\{04,20,40,60\}$
$4.5 !$
fav $=8(5 !-4 !)+4.5 !$
fav $=32.4 !+20.4 !=52.4 !$
$P =\frac{\text { fav }}{\text { Total }}=\frac{52.4 !}{180.4 !}=\frac{13}{45}$
Standard 11
Mathematics
