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  • From a class of 25 students, 10 are to be chosen for an excursion party. There are 3 students who decide that either all of them will join or none of them will join. In how many ways can they be chosen?

From a class of 25 students, 10 are to be chosen for an excursion party. There are 3 students who decide that either all of them will join or none of them will join. In how many ways can they be chosen?

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\\ All\ 3 \ join \\ Remaining \ 7 \ to \ be \ chosen\ from \ 25-3=22 \ students\\ Number \ of \ ways \ =^{22} C _{7} \\ \begin{array}{l} \\ =\frac{22 !}{7 !(22-7) !} \\ \\ =\frac{22 !}{7 !(15) !}=170544 \end{array}

\\ All \ 3 \ don't\ join\ \\ Remaining \ 10 \ chosen\ from \ 22 \ students \\ Number \ of \ ways \ =^{22} C _{10} \begin{array}{l} \\ =\frac{22 !}{10 !(22-10) !} \\ \\ =\frac{22 !}{10 ! 12 !}=646646 \end{array}

\\ Thus, \\ Total \ number \ of \ ways \ $=170544+646646$ =817190

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Deependra Verma

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