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There are five bags each containing identical sets of ten distinct chocolates. One chocolate is picked from each bag.

The probability that at least two chocolates are identical is __________

  1. $0.3024$
  2. $0.4235$
  3. $0.6976$
  4. $0.8125$
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Option C

$P(\text{No two chocolates are identical}) = \frac{10\times9\times8\times7\times6}{{10}^5} = \frac{30240}{{10}^5} = 0.3024$

$P(\text{At least two chocolates are identical}) = 1 – P(\text{No two chocolates are identical})$

$\qquad \qquad = 1 – 0.3024 = 0.6976$


Alternatively, 

Number of ways of selecting $5$ distinct chocolates, one each from the $5$ bags is same as selecting $5$ chocolates from $10$ distinct ones $ = {}^{10}C_5.$

If “distinct” requirement is not there, each of the $5$ chocolate has $10$ options $\implies 10^5.$

So, probability that no two chocolates are identical $ = \dfrac{{}^{10}C_5}{10^5} = 0.3024$

Probability that at least $2$ chocolcates are identical $ = 1 – 0.3024 = 0.6976$

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