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28 Feb 2024, 02:30
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C
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Find the number of natural numbers which lie between 10^7 and 10^8 and which have products of their digits as 6?
A. 36
B. 55
C. 64
D. 81
E. 105
A. 36
B. 55
C. 64
D. 81
E. 105
28 Feb 2024, 05:12
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Option C should be correct
\(10^{7}\) = 10000000 , so, we need 8 digit numbers which has 6 as the product of their digits.
6 = 2x3
6 = 1x6
we can have S1 = {1,1,1,1,1,1,2,3} and S2 = {1,1,1,1,1,1,1,6}
We can arrange the dogits of S1 in \(\frac{8!}{6!}\) = 7x8 = 56
We can arrange the dogits of S2 in \(\frac{8!}{7!}\) = 8x1 = 8
So, total number of natural numbers = (56+8) = 64
Hence, option C
\(10^{7}\) = 10000000 , so, we need 8 digit numbers which has 6 as the product of their digits.
6 = 2x3
6 = 1x6
we can have S1 = {1,1,1,1,1,1,2,3} and S2 = {1,1,1,1,1,1,1,6}
We can arrange the dogits of S1 in \(\frac{8!}{6!}\) = 7x8 = 56
We can arrange the dogits of S2 in \(\frac{8!}{7!}\) = 8x1 = 8
So, total number of natural numbers = (56+8) = 64
Hence, option C
28 Feb 2024, 09:04
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Find the number of natural numbers which lie between 10^7 and 10^8 and which have products of their digits as 6?
A. 36
B. 55
C. 64
D. 81
E. 105
10^7 means eight digits in the number.
Thus, ways in which 6 is product of the eitht digits are
A. Only digits 1,2 and 3 are used
A1. 1 2 3 _ _ _ _ _ = 6 numbers wherein the blanks are filled with 1 and 3 can take 6 places.
A2. 1 3 2 _ _ _ _ _ = 6 numbers wherein the blanks are filled with 1 and 2 can take 6 places.
A3. 2 1 3 _ _ _ _ _ = 6 numbers wherein the blanks are filled with 1 and 3 can take 6 places.
A4. 3 1 2 _ _ _ _ _ = 6 numbers wherein the blanks are filled with 1 and 2 can take 6 places.
A5. 1 1 2 3 _ _ _ _ = 5 numbers wherein the blanks are filled with 1 and 3 can take 5 places.
A6. 1 1 3 2 _ _ _ _ = 5 numbers wherein the blanks are filled with 1 and 2 can take 5 places.
A7. 1 1 1 2 3 _ _ _ = 4 numbers wherein the blanks are filled with 1 and 3 can take 4 places.
A8. 1 1 1 3 2 _ _ _ = 4 numbers wherein the blanks are filled with 1 and 2 can take 4 places.
A9. 1 1 1 1 2 3 _ _ = 3 numbers wherein the blanks are filled with 1 and 3 can take 3 places.
A10. 1 1 1 1 3 2 _ _ = 3 numbers wherein the blanks are filled with 1 and 2 can take 3 places.
A11. 1 1 1 1 1 2 3 _ = 2 numbers wherein the blanks are filled with 1 and 3 can take 2 places.
A12. 1 1 1 1 1 3 2 _ = 2 numbers wherein the blanks are filled with 1 and 2 can take 2 places
A13. 1 1 1 1 1 1 2 3 = 2 numbers wherein the 2 and 3 between each other i.e. 2 places.
A14. 2 3 1 _ _ _ _ _ = 1 number wherein the blanks are filled with 1 and 1 can take 6 places.
A15. 3 2 1 _ _ _ _ _ = 1 numbers wherein the blanks are filled with 1 and 1 can take 6 places.
Total = 56 numbers
B. Only digits 1 and 6 are used
6 _ _ _ _ _ _ _ = 8 numbers wherein 1 takes the rest of the blank places and 6 moves in those 8 places.
Total = 56 + 8 = 64 numbers
IMO Answer C.
A. 36
B. 55
C. 64
D. 81
E. 105
10^7 means eight digits in the number.
Thus, ways in which 6 is product of the eitht digits are
A. Only digits 1,2 and 3 are used
A1. 1 2 3 _ _ _ _ _ = 6 numbers wherein the blanks are filled with 1 and 3 can take 6 places.
A2. 1 3 2 _ _ _ _ _ = 6 numbers wherein the blanks are filled with 1 and 2 can take 6 places.
A3. 2 1 3 _ _ _ _ _ = 6 numbers wherein the blanks are filled with 1 and 3 can take 6 places.
A4. 3 1 2 _ _ _ _ _ = 6 numbers wherein the blanks are filled with 1 and 2 can take 6 places.
A5. 1 1 2 3 _ _ _ _ = 5 numbers wherein the blanks are filled with 1 and 3 can take 5 places.
A6. 1 1 3 2 _ _ _ _ = 5 numbers wherein the blanks are filled with 1 and 2 can take 5 places.
A7. 1 1 1 2 3 _ _ _ = 4 numbers wherein the blanks are filled with 1 and 3 can take 4 places.
A8. 1 1 1 3 2 _ _ _ = 4 numbers wherein the blanks are filled with 1 and 2 can take 4 places.
A9. 1 1 1 1 2 3 _ _ = 3 numbers wherein the blanks are filled with 1 and 3 can take 3 places.
A10. 1 1 1 1 3 2 _ _ = 3 numbers wherein the blanks are filled with 1 and 2 can take 3 places.
A11. 1 1 1 1 1 2 3 _ = 2 numbers wherein the blanks are filled with 1 and 3 can take 2 places.
A12. 1 1 1 1 1 3 2 _ = 2 numbers wherein the blanks are filled with 1 and 2 can take 2 places
A13. 1 1 1 1 1 1 2 3 = 2 numbers wherein the 2 and 3 between each other i.e. 2 places.
A14. 2 3 1 _ _ _ _ _ = 1 number wherein the blanks are filled with 1 and 1 can take 6 places.
A15. 3 2 1 _ _ _ _ _ = 1 numbers wherein the blanks are filled with 1 and 1 can take 6 places.
Total = 56 numbers
B. Only digits 1 and 6 are used
6 _ _ _ _ _ _ _ = 8 numbers wherein 1 takes the rest of the blank places and 6 moves in those 8 places.
Total = 56 + 8 = 64 numbers
IMO Answer C.
