Verified Solution • 2025–26 Session moderate

QJEEM22Ma7fe4: JEE Main 2022 Mathematics - Class 11 JEEM22Ma7fe4 Solution: Let be the set of all passwords which are six to eight…

Let SS be the set of all passwords which are six to eight characters long, where each character is either an alphabet from {A,B,C,D,E}\{A, B, C, D, E\} or a number from {1,2,3,4,5}\{1,2,3,4,5\} with the repetition of characters allowed. If the number of passwords in SS whose at least one character is a number from {1,2,3,4,5}\{1,2,3,4,5\} is α×56\alpha \times 5^{6} , then α\alpha is equal to ___________.

Step-by-Step Verified Solution

If password is 6 character long, then

Total number of ways having atleast one number =106−56= {10^6} - {5^6}

Similarly, if 7 character long =107−57= {10^7} - {5^7}

and if 8-character long =108−58= {10^8} - {5^8}

Number of password =(106+107+108)−(56+57+58)= ({10^6} + {10^7} + {10^8}) - ({5^6} + {5^7} + {5^8})

=56(26+5.27+25.28−1−5−25)= {5^6}({2^6} + {5.2^7} + {25.2^8} - 1 - 5 - 25)

=56(64+640+6400−31)= {5^6}(64 + 640 + 6400 - 31)

=7073×56= 7073 \times {5^6}

∴\therefore α=7073\alpha = 7073 .

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