Question 1135369: Triplet X,Y,Z are chosen from the set {1,2,3,.....24,25} such that X grater equal to Y and Y equal to Z.How many such triplets are possible?
Answer by ikleyn(52781)   (Show Source):
You can put this solution on YOUR website! .
Since in each triplet (X,Y,Z), the values Y and Z are equal, according to the condition,
you may forget about triplets and reformulate your problem EQUIVALENTLY in this much simpler way:
Doublets (X,Y) are chosen from the set {1,2,3,.....24,25} such that
X >= Y. How many such doublets are possible?
Solution
For Y = 1, there are 25 such doublets (1,1), (2,1), (3,1), . . . , (25,1).
For Y = 2, there are 24 such doublets (2,2), (3,2), . . . , (25,2).
For Y = 3, there are 23 such doublets (3,3), . . . , (25,3).
. . . . . . . . And so on . . . . .
Finally, for Y = 25, there is ONLY ONE such doublet (25,25).
So, the answer to your question is the value of the sum 25 + 24 + 23 + . . . + 1.
This value is very well known: it is the sum of the arithmetic progression
1 + 2 + 3 + . . . + 25 =  = 325.
ANSWER. The number of such triplets is 325.
Solved.
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