How many ways can 10 persons shake hands with two persons?

The question asks: How many different handshakes can happen between 10 people?

When Person A shakes hands with Person B, it's the same handshake as when Person B shakes hands with Person A. Since order doesn't matter, combinations are used.

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The number of ways to select 2 persons from 10 persons is given by $C(10, 2)$ or $^{10}C_2$.

The combination formula is:

$C(n, r) = \dfrac{n!}{r! \times (n-r)!}$

Where $n = 10$ (total number of people) and $r = 2$ (people selected for handshake).

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$C(10, 2) = \dfrac{10!}{2! \times (10-2)!}$

$C(10, 2) = \dfrac{10!}{2! \times 8!}$

$C(10, 2) = \dfrac{10 \times 9}{2 \times 1}$

$C(10, 2) = \dfrac{90}{2}$

$C(10, 2) = 45$

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Alternatively, the first person can shake hands with 9 others, the second person can shake hands with 8 remaining others, the third person with 7 remaining others, and so on.

Total $= 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45$

Therefore, the answer is 45.