Which of the following statements are true about the males in a colony of honey bees ? (i) They have 16 chromosomes per cell. (ii) They produce sperms by meiosis. (iii) They have a grandfather but no father. (iv) All males in the colony are haploid except one.

The correct option is (i) and (iii) only.

Let us break down the statements step-by-step to understand why they are true or false:

1. Chromosome number in male honey bees (drones):
Honey bees exhibit a unique sex-determination system called the haplodiploid system. In this system, females (queens and worker bees) are diploid, meaning they develop from fertilized eggs and have 32 chromosomes per cell. Male honey bees (drones), on the other hand, develop parthenogenetically from unfertilized eggs. As a result, males are haploid and contain only half the number of chromosomes.
Thus, male honey bees have:
$$\frac{32}{2}=16$$
chromosomes per cell. This makes statement (i) true.

2. Mode of sperm production:
Since male honey bees are already haploid (having 16 chromosomes), they cannot undergo meiosis (reduction division) to produce sperm cells, as this would halve their genetic material to 8 chromosomes. Instead, they produce sperm cells through mitosis (equational division) so that each sperm carries the full haploid set of 16 chromosomes. Therefore, statement (ii) is false.

3. Parentage of male honey bees (no father, but a grandfather):
Because male honey bees develop from unfertilized eggs laid by the queen bee (mother), they do not have a father. However, the queen bee itself developed from a fertilized egg, which means she had a mother (the grandmother) and a father (the grandfather). Therefore, the male drone inherits genetic material from its mother's father. Thus, a male honey bee has a grandfather but no father. This makes statement (iii) true.

4. Ploidy of males in the colony:
All male honey bees (drones) in a colony develop from unfertilized eggs and are haploid. There are no diploid males in a typical healthy colony. Therefore, statement (iv) is false.

In conclusion, only statements (i) and (iii) are correct, matching the correct option.