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 February 16th, 2014, 10:53 PM #1 Senior Member   Joined: Feb 2014 Posts: 114 Thanks: 1 Four-digit number Hi, How many four-digit number which each digit smaller than previous digit?
 February 17th, 2014, 12:04 AM #2 Math Team   Joined: Apr 2010 Posts: 2,780 Thanks: 361 Re: Four-digit number ${10 \choose 4}= 210$
 February 17th, 2014, 12:21 AM #3 Senior Member   Joined: Feb 2014 Posts: 114 Thanks: 1 Re: Four-digit number I don't it correct. Because each digit smaller than previous digit
 February 17th, 2014, 12:42 AM #4 Math Team   Joined: Apr 2010 Posts: 2,780 Thanks: 361 Re: Four-digit number Code: 1 9876 2 9875 3 9874 4 9873 5 9872 6 9871 7 9870 8 9865 9 9864 10 9863 11 9862 12 9861 13 9860 14 9854 15 9853 16 9852 17 9851 18 9850 19 9843 20 9842 21 9841 22 9840 23 9832 24 9831 25 9830 26 9821 27 9820 28 9810 29 9765 30 9764 31 9763 32 9762 33 9761 34 9760 35 9754 36 9753 37 9752 38 9751 39 9750 40 9743 41 9742 42 9741 43 9740 44 9732 45 9731 46 9730 47 9721 48 9720 49 9710 50 9654 51 9653 52 9652 53 9651 54 9650 55 9643 56 9642 57 9641 58 9640 59 9632 60 9631 61 9630 62 9621 63 9620 64 9610 65 9543 66 9542 67 9541 68 9540 69 9532 70 9531 71 9530 72 9521 73 9520 74 9510 75 9432 76 9431 77 9430 78 9421 79 9420 80 9410 81 9321 82 9320 83 9310 84 9210 85 8765 86 8764 87 8763 88 8762 89 8761 90 8760 91 8754 92 8753 93 8752 94 8751 95 8750 96 8743 97 8742 98 8741 99 8740 100 8732 101 8731 102 8730 103 8721 104 8720 105 8710 106 8654 107 8653 108 8652 109 8651 110 8650 111 8643 112 8642 113 8641 114 8640 115 8632 116 8631 117 8630 118 8621 119 8620 120 8610 121 8543 122 8542 123 8541 124 8540 125 8532 126 8531 127 8530 128 8521 129 8520 130 8510 131 8432 132 8431 133 8430 134 8421 135 8420 136 8410 137 8321 138 8320 139 8310 140 8210 141 7654 142 7653 143 7652 144 7651 145 7650 146 7643 147 7642 148 7641 149 7640 150 7632 151 7631 152 7630 153 7621 154 7620 155 7610 156 7543 157 7542 158 7541 159 7540 160 7532 161 7531 162 7530 163 7521 164 7520 165 7510 166 7432 167 7431 168 7430 169 7421 170 7420 171 7410 172 7321 173 7320 174 7310 175 7210 176 6543 177 6542 178 6541 179 6540 180 6532 181 6531 182 6530 183 6521 184 6520 185 6510 186 6432 187 6431 188 6430 189 6421 190 6420 191 6410 192 6321 193 6320 194 6310 195 6210 196 5432 197 5431 198 5430 199 5421 200 5420 201 5410 202 5321 203 5320 204 5310 205 5210 206 4321 207 4320 208 4310 209 4210 210 3210
February 19th, 2014, 07:37 PM   #5
Senior Member

Joined: Sep 2012
From: British Columbia, Canada

Posts: 764
Thanks: 53

Re: Four-digit number

Quote:
 Originally Posted by life24 I don't it correct. Because each digit smaller than previous digit
If you pick 4 digits out of the possible 10, then there is only one way to arrange them so that the digits are in decreasing order. The number of possible ways to do so is 10C4 = 210, hence 210 is your answer.

February 19th, 2014, 08:34 PM   #6
Math Team

Joined: Dec 2006
From: Lexington, MA

Posts: 3,267
Thanks: 408

Re: Four-digit number

Hello, life24!

Quote:
 How many four-digit numbers in which each digit smaller than previous digit?

$\text{Write the digits in descending order: }\;9\;8\;7\;6\;5\;4\;3\;2\;1\;0$

$\text{Now select any four digits: }\:{10\choose4} \,=\,120\text{ ways.}$

$\text{Do you see it?}$

 February 20th, 2014, 03:47 AM #7 Math Team   Joined: Apr 2010 Posts: 2,780 Thanks: 361 Re: Four-digit number He reposted the Question here. (10 C 4 = 210 ? 120)
 February 20th, 2014, 06:16 AM #8 Senior Member   Joined: Feb 2014 Posts: 114 Thanks: 1 Re: Four-digit number Hi, The final answer is 714
 February 20th, 2014, 10:16 AM #9 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Four-digit number Ah. So you didn't actually want the digits to be smaller than the last, just not greater. So 4444 would be allowed.

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