**Genotypes Controlled by 4 Types of Alleles**

In order to determine the number of possible genotypes formed by a character controlled by 4 types of alleles, we need to consider the principles of Mendelian genetics and the combinations of these alleles.

**Understanding Alleles:**
Alleles are alternate forms of a gene that can occupy the same position or locus on homologous chromosomes. They can be dominant or recessive, and individuals usually have two alleles for each gene, one from each parent.

**Types of Alleles:**
Since the character in question is controlled by 4 types of alleles, let's label them as A, B, C, and D.

**Possible Genotypes:**
To determine the number of possible genotypes, we need to consider the number of combinations that can be formed by these alleles.

- **Combining Two Alleles:**
For each gene, an individual can have two alleles. Therefore, we need to calculate the combinations when selecting two alleles out of four types.

The number of combinations of selecting two alleles out of four is calculated using the formula for combinations:

C(n, r) = n! / (r! * (n-r)!)

where n is the total number of items and r is the number of items to be selected.

C(4, 2) = 4! / (2! * (4-2)!)
C(4, 2) = 4! / (2! * 2!)
C(4, 2) = (4 * 3 * 2 * 1) / ((2 * 1) * (2 * 1))
C(4, 2) = 24 / (4 * 2)
C(4, 2) = 24 / 8
C(4, 2) = 3

Therefore, there are 3 different combinations when selecting two alleles out of four.

- **Combining Three Alleles:**
Similarly, we can calculate the number of combinations when selecting three alleles out of four.

C(4, 3) = 4! / (3! * (4-3)!)
C(4, 3) = 4! / (3! * 1!)
C(4, 3) = (4 * 3 * 2 * 1) / ((3 * 2 * 1) * (1))
C(4, 3) = 24 / (6 * 1)
C(4, 3) = 24 / 6
C(4, 3) = 4

So, there are 4 different combinations when selecting three alleles out of four.

- **Combining Four Alleles:**
Lastly, there is only one possible combination when selecting all four alleles.

C(4, 4) = 4! / (4! * (4-4)!)
C(4, 4) = 4! / (4! * 0!)
C(4, 4) = (4 * 3 * 2 * 1) / ((4 * 3 * 2 * 1) * (1))
C(4, 4) = 24 / (24 * 1)
C(4, 4) = 24 / 24
C(4, 4) = 1

Therefore, there is

By the formula,
The number of genotypes possible is the sum of integers between 1 and n.
n = number of given allels.
Here, n=4
Number of possible genotypes = 1+2+3+4=10
Hence B) is the right one.