Q81 / 150 · MHT CET 2023 (09 May Shift 1) · Mathematics

If at the end of certain meeting, everyone had shaken hands with everyone else, it was found that 45 handshakes were exchanged, then the number of members present at the meeting, are

  • Correct answer
    A
    10
  • B
    15
  • C
    20
  • D
    21

Solution

Let ' $n$ ' be the number of members in the meeting $\therefore \quad$ Total number of handshakes $={ }^{\mathrm{n}} \mathrm{C}_2$ $\therefore \quad{ }^{\mathrm{n}} \mathrm{C}_2=45$ $\begin{array}{ll} & \frac{\mathrm{n} !}{2 !(\mathrm{n}-2) !}=45 \\ & \frac{\mathrm{n}(\mathrm{n}-1)(\mathrm{n}-2) !}{2 \times(\mathrm{n}-2) !}=45 \\ & \mathrm{n}(\mathrm{n}-1)=90 \\ \therefore \quad & \mathrm{n}^2-\mathrm{n}-90=0 \\ & \mathrm{n}=10 \text { or } \mathrm{n}=-9 \text { (not possible) } \\ \therefore \quad & \mathrm{n}=10\end{array}$

Hence, the correct answer is option: 'A'.

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