One nurse missed the doctor's instructions with the medication she has to give to $10$ patients. $4$ patients are taking a pill for hypertension, $3$ patients are taking a pill for diabetes, $2$ patients are taking a pill for heart arrhythmia and $1$ patient is taking a pill for hypotension. The nurse randomly gives the pills to the patients. Coincidentally, everyone receives medication in the right way. What is the probability that this will happen?

I think that all the cases are $\dfrac{10!}{4! \cdot 3! \cdot 2! \cdot 1!}$ but I face difficulty in finding the sum of cases that each patient takes the right medicine.I would appreciate for your quidance. Thank you very much in advance.

  • $\begingroup$ Consider permutations of $(a,a,b)$ amongst three patients. The number of cases is $\frac{3!}{2!}=3$: $\{(a,a,b),(a,b,a),(b,a,a)\}$. How many are correct? is it not just 1? $\endgroup$ – Rahul Madhavan Apr 17 at 17:27
  • $\begingroup$ You helped me a lot, Thank you very much! $\endgroup$ – Magda Apr 17 at 19:34

Suppose the patients are given the pills in the order $H_1H_2H_3H_4D_1D_2D_3A_1A_2O_1$ where $H_i$ represents the person who has to take the pill for hypertension, $D_i$ for diabetes, $A_i$ for arrhythmia and $O_1$ for hypotension.

Now, probability that $H_1$ gets the right pill is $4/10$, then $H_2$ gets the right pill is $3/9$ so in this way, the probability that all get the right pill would be $$\dfrac{4}{10} \cdot \dfrac{3}{9} \cdot \dfrac{2}{8} \cdot \dfrac{1}{7} \cdot \dfrac{3}{6} \cdot \dfrac{2}{5} \cdot \dfrac{1}{4} \cdot \dfrac23 \cdot \dfrac12 \cdot 1 $$

Note that the order in which they are given the pills could be anything, probability will still remain the same.

  • $\begingroup$ Thank you very much, really helpful! $\endgroup$ – Magda Apr 17 at 19:34
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    $\begingroup$ @Magda If you found it useful, feel free to check mark my answer and give an upvote :) $\endgroup$ – Light Yagami Apr 17 at 19:35
  • $\begingroup$ My pleasure! Thank you very much you helped me fully! $\endgroup$ – Magda Apr 17 at 20:14

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