What is the number of length-7 sequences from the numerical digits {0,1,2,3,4,5,6,7,8,9} such that the sequence uses exactly 4 different digits?

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Keely Fernandez · Accepted Answer

1) Choose exactly 4 different digits from the list of 10 digits.2) Arrange above 4 different digits at 7 places such that the created sequence consists of exactly 4 different digits.Find the number of ways for doing the first point.From the selection or combination rule it will be       10                  C        4              .. Evaluate       10                  C        4              . using the formula       n                  C        r              =            n      !              r      !      (      n      −      r      )      !            10                  C        4              =            10      !              4      !      (      10      −      4      )      !        =            10      !              4      !      ×      6      !        =            10      ×      9      ×      8      ×      7      ×      6      !              24      ×      6      !        =  10  ×            9      ×      8        24    ×  7  =  10  ×  3  ×  7  =  210Thus, the first event done in 210 waysNow, find the number of ways for doing a second event. Use the above four numbers to create a length-7 sequence without any condition.Thus, way each digit has 4 choices therefore the total number of such sequences is 47.We have to make a sequence which consists of exactly 4 different digits. Remove the sequences from a set which has only 3 different digits or less than it.Find the number of sequences which consist of 3 different digits or less than it from the selected 4 different digits. Then choose any three digits from 4 different digits by       4                  C        3             ways. Each position has 3 choices and there are 7 such positions so numbers of choices is 37.Therefore, such  sequences are        4                  C        3              ×      3    7  .Subtract       4                  C        3              ×      3    7   from       4    7    to get the number of sequences which consist of exactly 4 different digits which are selected in the first step.      4    7    −      4                  C        3              ×      3    7    =  16384  −  4  (  2187  )  =  16384  −  8748  =  7636Thus, the second event done in 7636 waysUse the fundamental principle of counting to find required the number of length-7 sequences. Since number of ways of occurring first event is 210 ways and second event is 7636 ways so required number is   =  210  ×  7636   =  1603560

What is the number of length-7 sequences from the numerical

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