Question

how many 2-letter combinations can be created from the distinct letters in the word \mathematician\? assume that the order of the letters does not matter.
28
56
78
156

Explanation:

Step1: Identify distinct letters

First, list distinct letters in "mathematician": m, a, t, h, e, i, c, n. Total distinct letters $n=8$.

Step2: Apply combination formula

We use the combination formula for choosing $k=2$ letters from $n=8$, where order does not matter:
$$C(n,k)=\frac{n!}{k!(n-k)!}$$
Substitute $n=8$, $k=2$:
$$C(8,2)=\frac{8!}{2!(8-2)!}=\frac{8\times7\times6!}{2\times1\times6!}$$

Step3: Simplify the expression

Cancel out $6!$ in numerator and denominator:
$$\frac{8\times7}{2\times1}=\frac{56}{2}=28$$

Answer:

28