Answer :

Answer:

32767 distinct color grouping

Step-by-step explanation:

Given data:

Total tube is 13

for distinct color, group can be 1,2,3,4,.... 15

For total number of ways we have following relation

[tex]^nC_0 +^nC_1 +^nC_2 + .......^nC_n = 2^n[/tex]

here n = 13 so we have

[tex]^15C_0 +^15C_1 +^15C_2 + .......^15C_{15} = 2^15[/tex]

for at least one

[tex]= 2^{15} - 1[/tex]

[tex]= 32768 -1[/tex]

= 32767 distinct color grouping

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