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  • The number of arrangements of all digits of 12345 such that at least 3 digits will not come in its position isOption: 1 89<div class='q

The number of arrangements of all digits of 12345 such that at least 3 digits will not come in its position is

Option: 1

89


Option: 2

109


Option: 3

78


Option: 4

57


Answers (1)

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Total number of ways such that at least 3 digits will not come in its position.

=Derangement of 3 digits +Derangement of 4 digits +Derangement of 5 digits

\begin{aligned} & ={ }^5 \mathrm{C}_3\left\{3 !-{ }^3 \mathrm{C}_1 2 !+{ }^3 \mathrm{C}_2 1 !-{ }^3 \mathrm{C}_3 0 !\right\} \\ \\& +{ }^5 \mathrm{C}_4\left[4 !-{ }^4 \mathrm{C}_1(3 !)+{ }^4 \mathrm{C}_2(2 !)-{ }^4 \mathrm{C}_3(1 !)+{ }^4 \mathrm{C}_4(0 !)\right\} \\ \\& \quad+{ }^5 \mathrm{C}_5\left\{5 !-{ }^5 \mathrm{C}_1 4 !+{ }^5 \mathrm{C}_2 3 !-{ }^5 \mathrm{C}_3 2 !+{ }^5 \mathrm{C}_4 1 !-{ }^5 \mathrm{C}_5(0 !)\right\} \\ \\& =10(2)+5(9)+(44) \\ \\& =20+45+44=109 \end{aligned}

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