dancing with the stars elimination
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We use generating functions. The generating function for one day is:
So the number of favorable sequences is $60691$.
This fraction is already in lowest terms (can be verified via GCD computation), so the final answer is:
Question: A university professor assigns a genetics research project where students must simulate genetic drift using 6 alleles of varying frequencies. If each allele is modeled as being passed independently with probability proportional to its frequency, and each simulation run selects one allele uniformly based on relative frequency, what is the probability that in 10 consecutive simulations, exactly 3 distinct alleles appear?
We are to compute the probability that in 10 independent trials (simulations), exactly 3 distinct alleles among 6 are observed, where at each step, an allele is selected with probability proportional to its frequency (i.e., multinomial sampling without replacement — but since no actual frequencies are given, assume uniform distribution over alleles for symmetry). However, the problem states selected uniformly based on relative frequency, but without specific weights, and to make it solvable, we 
We count the number of sequences of length 10 with exactly 3 distinct alleles.
Step 2: Count the number of surjective functions from 10 trials to these 3 alleles — i.e., all 3 appear at least once.
Divide numerator and denominator by 16:
Check GCD: Suppose no further reduction — use decimal approximation for verification:
Reduce by dividing numerator and denominator by 24 (GCD computation yields):