The total number of natural numbers of six di | vedclass

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Question

(A) There can be two types of numbers formed using the digits $1, 2, 3, 4$ such that each appears at least once in a $6$-digit number:$(i)$ One digit

Vedclass · Accepted Answer

$1560$

Vedclass · Answer

(A) There can be two types of numbers formed using the digits $1, 2, 3, 4$ such that each appears at least once in a $6$-digit number:$(i)$ One digit repeats $3$ times and the other three digits appear once each (e.g.,$1, 1, 1, 2, 3, 4$).The number of such arrangements is given by $\frac{6!}{3!1!1!1!} 	imes \binom{4}{1} = \frac{720}{6} 	imes 4 = 120 	imes 4 = 480$.$(ii)$ Two digits repeat $2$ times each and the other two digits appear once each (e.g.,$1, 1, 2, 2, 3, 4$).The number of such arrangements is given by $\frac{6!}{2!2!1!1!} 	imes \binom{4}{2} = \frac{720}{4} 	imes 6 = 180 	imes 6 = 1080$.Therefore,the total number of such $6$-digit numbers is $480 + 1080 = 1560$.

The total number of natural numbers of six digits that can be made with digits $1, 2, 3, 4$,if all digits are to appear in the same number at least once,is

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