Permutation and Combination Questions

Question Number 101880 by mr W last updated on 05/Jul/20

$${Find}\:{the}\:{number}\:{of}\:{six}−{digit}\:{odd} \\$$$${numbers}\:{without}\:{repeated}\:{digits}. \\$$

Answered by bemath last updated on 05/Jul/20

$$\mathrm{5}×\mathrm{8}×{C}_{\mathrm{4}} ^{\mathrm{8}} ×\mathrm{4}! \\$$

Commented by mr W last updated on 05/Jul/20

$${correct},\:{thanks}! \\$$

Answered by som(math1967) last updated on 05/Jul/20

$$\mathrm{Odd}\:\mathrm{numbers}\:\mathrm{unitdigit}\:\mathrm{1} \\$$$$\overset{\mathrm{9}} {\mathrm{p}}_{\mathrm{5}} −\overset{\mathrm{8}} {\:}\mathrm{p}_{\mathrm{4}} =\mathrm{9}×\mathrm{8}×\mathrm{7}×\mathrm{6}×\mathrm{5}−\mathrm{8}×\mathrm{7}×\mathrm{6}×\mathrm{5} \\$$$$=\mathrm{8}×\mathrm{7}×\mathrm{6}×\mathrm{5}\left(\mathrm{9}−\mathrm{1}\right) \\$$$$=\mathrm{8}×\mathrm{7}×\mathrm{6}×\mathrm{40} \\$$$$\therefore\mathrm{total}\:\mathrm{odd}\:\mathrm{numbers}=\mathrm{56}×\mathrm{240}×\mathrm{5} \\$$$$=\mathrm{67200} \\$$

Commented by som(math1967) last updated on 05/Jul/20

$$\mathrm{welcome}\:\mathrm{sir} \\$$

Commented by mr W last updated on 05/Jul/20

$${correct},\:{thanks}! \\$$