Answer
The ordered arrangements:
$123,124,125,126,134,132,135,136,142,143,145,146,142,153,154,156,162,163,164,165$
$213,214,215,216,231,234,235,236,241,243,245,246,251,253,254,256,261,263,264,265$
$312,314,315,316,321,324,325,326,341,342,345,346,351,352,354,356,361,362,364,365$
$412,413,415,416,421,423,425,426,431,432,435,436,451,452,453,456,461,462,463,465$
$512,513,514,516,521,523,524,526,531,532,534,536,541,542,543,546,561,562,563,564$
$612,613,614,615,621,623,624,625,631,632,634,635,641,642,643,645,651,652,653,654$
$P(6,3)=120$.
Work Step by Step
The ordered arrangements:
$123,124,125,126,134,132,135,136,142,143,145,146,142,153,154,156,162,163,164,165$
$213,214,215,216,231,234,235,236,241,243,245,246,251,253,254,256,261,263,264,265$
$312,314,315,316,321,324,325,326,341,342,345,346,351,352,354,356,361,362,364,365$
$412,413,415,416,421,423,425,426,431,432,435,436,451,452,453,456,461,462,463,465$
$512,513,514,516,521,523,524,526,531,532,534,536,541,542,543,546,561,562,563,564$
$612,613,614,615,621,623,624,625,631,632,634,635,641,642,643,645,651,652,653,654$
We know that $P(n,r)=n(n-1)(n-2)...(n-k+1).$ Also $P(n,0)=1$ by convention.
Hence $P(6,3)=6\cdot5\cdot4=120$.
