CUET UG 2025 — General Test Quantitative Reasoning

When each player shakes hands with every other player exactly once, the total number of handshakes can be calculated using the combination formula.

For $n$n players, each player shakes hands with $(n-1)$(n−1) other players. However, this counts each handshake twice (once for each person involved), so the total is divided by 2.

Number of handshakes $= \dfrac{n(n-1)}{2}$=2n(n−1)​

---

Given that the total number of handshakes is 120:

$$\dfrac{n(n-1)}{2} = 120$$2n(n−1)​=120

$$n(n-1) = 240$$n(n−1)=240

$$n^2 - n = 240$$n2−n=240

$$n^2 - n - 240 = 0$$n2−n−240=0

---

Factoring the quadratic equation, we need two numbers that multiply to $-240$−240 and add to $-1$−1.

These numbers are $16$16 and $-15$−15.

$$(n - 16)(n + 15) = 0$$(n−16)(n+15)=0

$n = 16$n=16 or $n = -15$n=−15

---

Since the number of players must be positive, $n = 16$n=16.

Therefore, the number of players is 16.