Use a unique mode to bound p, then sketch a different distribution

Five people are sampled from a large village. (i) X counts those who are underweight; state in context an assumption needed for a binomial model. (ii) The village proportion underweight is p, and the mode of X is 2. Show that $1/3<p<1/2$. (iii) Given instead $X\sim B(5,0.25)$, sketch its distribution.