A student selects 3 digits from numbers 1 to 9 such that they are in strictly increasing order. How many selections have the property that the three digits form an arithmetic progression?
A. 7
B. 14
C. 12
D. 16

• Numbers can only be
  123,234,345,456,567,678,789 ( Common difference of 1)
  135,246,357,468,579 ( Common difference of 2)
  147,258,369 ( Common difference of 3)
  159 ( common difference of 4)
  Total 16 selections
  Option D
• 11 years agoHelpfull: Yes(58) No(0)
• answer will be 16 as we can select in 16 ways and i.e
  123
  234
  345
  456
  567
  678
  789
  135
  246
  357
  468
  579
  147
  258
  369
  159
• 11 years agoHelpfull: Yes(8) No(0)
• 16 is the ans
  
• 11 years agoHelpfull: Yes(6) No(1)
• 16
  123,234,345,567,678,789,246,468,
  258,147,369,135,357,579,456,159.
• 11 years agoHelpfull: Yes(3) No(0)
• it is 16.. (1,2,3),(2,3,4),(3,4,5),(4,5,6),(5,6,7),(6,7,8),(7,8,9),(2,4,6),(3,6,9),(1,3,5),(3,5,7),(5,7,9),(1,4,7),(1,5,9),(4,6,8),(2,5,8)
• 11 years agoHelpfull: Yes(2) No(0)
• 16 is the ans
  
• 11 years agoHelpfull: Yes(2) No(2)
• dioption d
  
• 11 years agoHelpfull: Yes(1) No(0)
• Let
  a
  =
  b
  −
  d
  
  Let
  b
  =
  b
  
  Let
  c
  =
  b
  +
  d
  
  Given that
  a
  ,
  b
  ,
  c
  must be distinct digits from the set
  {
  1
  ,
  2
  ,
  3
  ,
  4
  ,
  5
  ,
  6
  ,
  7
  ,
  8
  ,
  9
  }
  , we also have the conditions:
  
  a
  ≥
  1
  
  c
  ≤
  9
  
  From these conditions, we can derive the following inequalities:
  
  b
  −
  d
  ≥
  1
  implies
  b
  ≥
  d
  +
  1
  
  b
  +
  d
  ≤
  9
  implies
  b
  ≤
  9
  −
  d
  
  Combining these inequalities gives:
  
  d
  +
  1
  ≤
  b
  ≤
  9
  −
  d
  
  
  To ensure that
  b
  has valid values, we need
  9
  −
  d
  ≥
  d
  +
  1
  :
  
  9
  −
  d
  ≥
  d
  +
  1
  ⟹
  9
  −
  1
  ≥
  2
  d
  ⟹
  8
  ≥
  2
  d
  ⟹
  d
  ≤
  4
  
  
  Now we can consider the possible values of
  d
  from 1 to 4 and count the valid values of
  b
  for each
  d
  :
  
  For
  d
  =
  1
  :
  The range for
  b
  is
  2
  to
  8
  (inclusive).
  Possible values:
  2
  ,
  3
  ,
  4
  ,
  5
  ,
  6
  ,
  7
  ,
  8
  (7 values).
  For
  d
  =
  2
  :
  The range for
  b
  is
  3
  to
  7
  (inclusive).
  Possible values:
  3
  ,
  4
  ,
  5
  ,
  6
  ,
  7
  (5 values).
  For
  d
  =
  3
  :
  The range for
  b
  is
  4
  to
  6
  (inclusive).
  Possible values:
  4
  ,
  5
  ,
  6
  (3 values).
  For
  d
  =
  4
  :
  The range for
  b
  is
  5
  to
  5
  (inclusive).
  Possible value:
  5
  (1 value).
  Next, we sum the number of valid selections for each value of
  d
  :
  
  For
  d
  =
  1
  : 7 selections
  For
  d
  =
  2
  : 5 selections
  For
  d
  =
  3
  : 3 selections
  For
  d
  =
  4
  : 1 selection
  Calculating the total:
  
  7
  +
  5
  +
  3
  +
  1
  =
  16
  
  
  Thus, the total number of selections of 3 digits from the numbers 1 to 9 that form a strictly increasing arithmetic progression is
  
  16
• 2 Months agoHelpfull: Yes(0) No(0)