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If $7 \cdot ^nP_3 = 20 \cdot ^{n+1}P_2$,then $n = \dots$

All the letters of the word $ABRACADABRA$ are arranged in different possible ways. The number of such arrangements in which the vowels are together is:

${ }^{15} P_8 = A + 8 \cdot { }^{14} P_7 \Rightarrow A = $

Find $r$ if $^{5}P_{r} = ^{6}P_{r-1}$.

How many words,with or without meaning,each of $2$ vowels and $3$ consonants can be formed from the letters of the word $DAUGHTER$?

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