How many students does Harvard graduate each year? How many employees does Google have? If one year Google hired the entire math and CS graduating class of Harvard, would it still be "pretty rare" to see one of those people at Google?
I'm sure it's fine, but it's not Harvard or Stanford or MIT - it has a 45% acceptance rate similar to my school (~45-51%). AFAIK it's not even considered a public Ivy like UMich or UW or UNC.
Since you clearly like math, suppose there are 20 million students applying to college every year. One million of them are very bright (2-sigma intelligence). Suppose that every Ivy League accepts around two thousand students every year. Can the sum of Ivy Leagues fit all of the smart people?
Bonus question: What if we change our constraints to only account for 3-sigma people? Does our conclusion change?
Of course there's a distribution of those folks at all other universities, especially at top public ivies like GTech and UCB. But the cumulative effect of being at a lower tier institution can be pretty significant when searching for future opportunities. One thing I've noticed among people at my school is that even if they are very capable, they often don't apply to top internships, top programs for grad school etc, just because they don't see it as an option.
I think you made an astute observation: top people at lower ranked schools sometimes don't apply to top internships or grad schools because they don't see it as an option.
This speaks to their lack of confidence more than their capability. Perhaps that is one of the advantages a good school, a good peer group, or a good network can confer: the confidence to aim higher.
People don't think they're good enough... which may be true, but no one can truly know until they try. Self-limiting thinking is particularly prevalent in rust-belt cities and regions where knowledge or achievement is not prized, so people in knowledge-intensive fields have no models to emulate.
And sometimes when they try (it's not unusual for graduates of lower ranked universities to send out 300+ resumes only to get single digit responses), they get demoralized when they don't succeed on their first few tries, when in fact there's more than one path in life -- if one doesn't have natural advantages, one might have to embrace the more circuitous path(s). This can mean joining a startup, going to a better grad school than one's undergrad, moving to a better city to
upgrade one's peer groups (this is more important than most people think [1]), etc.
Life can surprise you if you keep trying and pivoting (ugh cliche, but there it is). There's an element of randomness and stochasticity in a free market, and I've seen enough counterexamples to distrust a static conception of how things "should be". (except for some stodgy areas like investment banking that only hire from certain schools; but even then there are backdoors)
* You're a new graduate, and the hardest hurdle you have to overcome is to get in the door. If you manage to do that and are able to prove yourself, your undergrad degree will become less and less and important. If you google Fortune 500 company CEOs, especially in non-tech companies, (you can do this exercise for yourself) you will learn that many of them went to non-elite schools for undergrad. For all its elite colleges, America is not really an academic-technocratic society (unlike countries like Germany where most CEOs have Ph.D.s). There are elements of William James' pragmatic philosophy that still influence the thinking in this country -- getting results is more important than academic knowledge.
Might be true for the university as a whole, but many of the colleges at Minnesota are quite selective. The College of Science and Engineering is one example, accepting 1177 out of 14,000 applications for 2017.
More context on the big differences in selectivity between colleges: Minnesota used to have "General College"[1] which, by design, admitted every student regardless of qualifications. That was changed in 2005, but the legacy of inclusion over selectivity lives on in some places.
I can say that CSE was very selective when I was there, and getting into upper division was even harder. But overall I don't think acceptance rate is a very useful statistic because program size affects it so much.
