Math worksheet generator

Answer key · worked solutions

1. 1. B. 603

   Roots-of-unity / residue-class balance on subset sums. Not $\binom{15}{5}/5$ because sums are not equally distributed mod $5$. $603$.

2. 2. E. 161

   Inclusion-exclusion after $y_i=x_i-1$. Upper bound 8. $161$.

3. 3. C. 756

   Inclusion–exclusion on $y_i=x_i-1$ with upper bounds. Shift to nonnegative $y_i$ summing to $12$, cap each $y_i\le5$. $756$.

4. 4. B. 28

   $\binom{N-1}{k-1}$. $\binom{8}{2}$. $28$.

5. 5. A. 9720

   $\binom{2n}{n}$ minus paths through the obstacle. Subtract product of binomials. $9720$.

6. 6. A. 2736

   Roots-of-unity filter / recurrence. Not exactly $2^n/3$. $2736$.

7. 7. B. 120

   $\binom{m}{k}$. $\binom{10}{3}$. $120$.

8. 8. B. 153

   $\binom{m}{k}$. $\binom{18}{2}$. $153$.

9. 9. C. 2207

   $\binom{2n}{n}$ minus paths through the obstacle. Subtract product of binomials. $2207$.

10. 10. E. 7970

    $\binom{2n}{n}$ minus paths through the obstacle. Subtract product of binomials. $7970$.