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Question: Let X= {S1,S2,S3,S4,S5} and Y = {L1,L2,L3} be the sets of satellites and launching sites respectively. If each site launches at least one satellite, find the number of ways to launch satellites in the space.

My solution:
L1 have choice of 5 satellites
L2 have choice of 4 satellites
L3 have choice of 3 satellites
Remaining two satellites can choose any station among 3
Hence my answer is 5*4*3*3*3. But the answer is wrong.

Pepper31
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  • It is wrong because you paid special importance to which of the satellites was the "first choice" satellite for each launching pad when that doesn't matter. Having picked $S_1$ for $L_1$, then picked $S_2$ for $L_2$, then $S_3$ for $L_3$ and then picked $L_1$ for both $S_4$ and $S_5$ is considered the same but counted separately than having picked $S_4$ for $L_1$, $S_2$ for $L_2$, then $S_3$ for $L_3$ and then picked $L_1$ for both $S_1$ and $S_5$. – JMoravitz Aug 02 '20 at 18:06
  • For a correct approach, use inclusion-exclusion or use stirling numbers of the second kind. – JMoravitz Aug 02 '20 at 18:06
  • See my answer here: https://math.stackexchange.com/questions/3768474/4-element-subset-of-1-6-that-includes-1-or-4-and-2-or-5-and/3768514#3768514 – YJT Aug 02 '20 at 18:09
  • @YJT did you paste the wrong link? That seems completely unrelated. – JMoravitz Aug 02 '20 at 18:11
  • It's a different question but the same common mistake to which I refer in bold. – YJT Aug 02 '20 at 18:30

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