Determine the number of 8-tuples ( ₁, ₂, , ₈ ) such that ₁, ₂, , ₈ 1,-1 and ₁+2 ₂+3 ₃+ +8 ₈ is a multiple of 3 .

Step-by-step solution

₁, ₂, ₃, . ₈ -1,1 . ₁+2 ₂+3 ₃+ +8 ₈ ( ₁+4 ₄+7 ₇ )+ (2 ₂+5 ₅+8 ₈ )+3 ( ₃+2 ₆ ) ( ₁+ ₄+ ₇ )+2 ( ₂+ ₅+ ₈ )+3 ~m ( ~m is an integer ) ( ₁- ₂ )+ ( ₄- ₅ )+ ( ₇- ₈ )+3 q ( q is an integer) ₁- ₂=2,0,-2 & similarly others p₁₂+p₄₅+p₇₈+3 q (where p ₁₂= ₁- ₂ , p₄₅= ₄- ₅ , . p ₇₈= ₇- ₈ )p₁₂=p₄₅=p₇₈=0 8 cases p₁₂=p₄₅=p₇₈=2( or -2 ) 2 cases p ₁₂=2, p ₄₅=-2, p ₇₈=0 (2 6) cases Hence number of tuples =22 4=88 ( ₃ & ₆ can be any one of 1 or -1 )