 Binomial count and iid values in fixed-budget rejection sampling

For n independent proposals and an envelope constant $M$, each rejection decision succeeds with probability $1/M$. The accepted count is Bin(n,1/M). Conditional on any acceptance pattern, the accepted values have independent target density f; summing over patterns preserves that product law conditional on the count. The empty-output probability is $(1-1/M{)}^n$.