A class has 100 students.Let $a_i$,1$\leq$i$\leq100$,denotes the number of friends the i-th student has in the class.For each 0$\leq$j$\leq$99,let $c_j$ denotes the number of students having at least j friends.Show that $\sum_{i=1}^{100}${$a_i$} =$\sum_{j=0}^{99}${$c_j$}
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This is your third question on the site, and for the third time you have posted a problem without showing any efforts yourself or giving any context. – Théophile Nov 24 '20 at 17:53
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What have you tried? Did you look at small cases? Do you have any ideas of why this statement is true? – Calvin Lin Nov 24 '20 at 17:54
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Check your summation indices again. E.g. The statement isn't true in the scenario that only 2 people are friends, in which case $a_ 1 = a_2 = 1$ and $ c_0 = c_1 = 2$. (Alternatively, $c_j$ is defined as a strict inequality) – Calvin Lin Nov 24 '20 at 17:56
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There's an issue with your summation indices. You likely want
$$ \sum_{i=0}^{100} a_i = \sum_{j=1}^{99} c_j $$
(Fill in the gaps as needed. If you're stuck, write out your working and thought process to demonstrate where you're at.)
Hint: If a person has exactly $k$ friends, then they have at least $1, 2, \ldots k$ friends.
Calvin Lin
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@mukeshmittal Your question was closed because you're required to show your work, not merely demand answers. You can edit the post with what you've tried and why you're stuck, and ask for it to be reopened. – Calvin Lin Dec 05 '20 at 01:43
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I have not solved this type of problem.I don't know how to proceed. – mukesh mittal Dec 05 '20 at 03:06