Question

find the number of 5 letter words that can be formed using the letters of the word CONSIDER. how many of them begin with C, how many end with R ? (combinations and permutations)

Answer 1

$Number \: of \: different \: letters \: in \:the \\word \: \blue {CONSIDER } = 8$

$The\: number\: of \:5 \:letter \:words\: that \\can\: be \:formed\: using \:the\: letters\: of\: given\\ word \: \blue {CONSIDER }= ^{8}P_{5}$

$Out \:of \:them ,$

$i ) If \: first \:place \:is \: filled \:by \: \red{ C} , \\</p><p>the \:remaining \: 4 \: placed \:by \:rest \:of \: 7 \\letters \:can \:be \: filled \: in \: ^{7}P_{4} \:ways$

$\therefore Number \:of \:words \: begin\: with \\\red{C} \:are \: ^{7}P_{4}$

$ii) If \: last \:place \:is \: filled \:by \: \red{ R} , \\</p><p>the \:remaining \: first \: 4 \: placed \:by \:rest \:of \\ 7 \\letters \:can \:be \: filled \: in \: ^{7}P_{4} \:ways$

$\therefore Number \:of \:words \: end \: with \\\red{R} \:are \: ^{7}P_{4}$

$iii ) If \: first \: place \:is \: filled \: with \:\blue {C} \\and \: last \:place \: is \: filled \:by \: \red{R}, \\</p><p>the \: remaining \:3 \: places \: between \: them \\by \: Remaining \: 6 \: letters \: can \:be \: filled \:in\\</p><p>^{6}P_{3} \:ways$

$\therefore Number \:of \:words \: begin \: with \\\red{C} \:and \:end \: with \: \red{R}\: are \: ^{6}P_{3}$

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