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  • If the number of ways of selecting n coupons out of an unlimited number of coupons bearing the letters A, T, C so that they can not be used to spell to the used CAT is 189 , then <im

If the number of ways of selecting n coupons out of an unlimited number of coupons bearing the letters A, T, C so that they can not be used to spell to the used CAT is 189 , then \Sigma n^2 must be

Option: 1

90


Option: 2

92


Option: 3

93


Option: 4

91


Answers (1)

best_answer

Number of ways selecting n coupons consisting of C or \mathrm{A}=2^n  \left(\begin{array}{l} \because \text { The word CAT can not be written } \\ \text { if at least one letters is not selected } \end{array}\right)

Now number of ways of selecting n coupons bearing only A=1^n.

\therefore Total number of ways =2^n+2^n+2^n-1^n-1^n-1^n

=3\left(2^n-1\right)

Given    3\left(2^n-1\right)=189

\begin{array}{rlrl} \Rightarrow & \: \: 2^n-1 & =63 \\ \\\Rightarrow &\: \: 2^n=64 & =2^6 \\ \\\therefore & \: \: n =6 \end{array}

then \sum n^2=\frac{n(n+1)(2 n+1)}{6}

                      =\frac{6 \cdot 7 \cdot 13}{6}=91

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Pankaj

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