Student Puzzle Editor Anirban DasGupta explains the solution:

For part (a), writing q for p2 = cp, the likelihood function is

((1 − p)p4q4)(p4q3(1 − q))((1 − q))(q3p2(1 − p))(p5q4(1 − q))

= p29(1 − p)2(1 − cp)3.

For part (b), the MLE of p is (32c+31)/68c , if it is not larger than 1.

For part (c), the likelihood function is a product of betas, (p(1−p))n/2(q(1−q))n/2.

The modal values converge to 1/2.