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Question Number 198283 by cortano12 last updated on 16/Oct/23
$$\mathrm{Given}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\:\mathrm{consisting} \\ $$$$\:\mathrm{of}\:\mathrm{4}\:\mathrm{digits}\:\mathrm{abcd}\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\:\mathrm{a}\leqslant\mathrm{b}\leqslant\mathrm{c}\leqslant\mathrm{d}\:\mathrm{is}\:... \\ $$$$\:\left(\mathrm{A}\right)\:\mathrm{495}\:\:\:\left(\mathrm{B}\right)\:\mathrm{385}\:\:\:\:\:\left(\mathrm{C}\right)\:\mathrm{275} \\ $$$$\:\left(\mathrm{D}\right)\:\mathrm{165}\:\:\:\:\left(\mathrm{E}\right)\:\mathrm{55}\: \\ $$
Commented by mr W last updated on 19/Oct/23
$$\left({A}\right)\:\mathrm{495} \\ $$$${generally}\:{with}\:{k}\:{digits}:\:\frac{\left({k}+\mathrm{8}\right)!}{\mathrm{8}!{k}!} \\ $$$${see}\:{Q}\mathrm{117061} \\ $$
Answered by MM42 last updated on 16/Oct/23
$$\left\{{a},{b},{c},{d}\right\}\:\rightarrow\:\begin{pmatrix}{\mathrm{9}}\\{\mathrm{4}}\end{pmatrix}=\mathrm{126} \\ $$$$\left\{{a},{a},{b},{c}\right\}\:\rightarrow\:\begin{pmatrix}{\mathrm{9}}\\{\mathrm{3}}\end{pmatrix}×\mathrm{3}=\mathrm{252} \\ $$$$\left\{{a},{a},{b},{b}\right\}\:\rightarrow\:\begin{pmatrix}{\mathrm{9}}\\{\mathrm{2}}\end{pmatrix}=\mathrm{36} \\ $$$$\left\{{a},{a},{a},{b}\right\}\:\rightarrow\:\begin{pmatrix}{\mathrm{9}}\\{\mathrm{2}}\end{pmatrix}×\mathrm{2}=\mathrm{72} \\ $$$$\left\{{a},{a},{a},{a}\right\}\:\rightarrow\mathrm{9}\: \\ $$$$ \\ $$$${ans}=\mathrm{495}\:\checkmark\:\rightarrow\:{A} \\ $$$$ \\ $$
Commented by cortano12 last updated on 17/Oct/23
$$\mathrm{why}\:\left(\mathrm{a},\mathrm{a},\mathrm{a},\mathrm{b}\right)=\begin{pmatrix}{\mathrm{9}}\\{\mathrm{2}}\end{pmatrix}\:×\mathrm{2}\:? \\ $$$$ \\ $$
Commented by mr W last updated on 17/Oct/23
$$\left({a},{b},{b},{b}\right)\:{and}\:\left({a},{a},{a},{b}\right) \\ $$
Commented by MM42 last updated on 17/Oct/23
$$\:\underline{\underbrace{\lesseqgtr}} \\ $$