A Universal Product Code (UPC) is a numerical representation of the bar codes used in stores, with an error-detecting feature to handle misscanned codes. A UPC is a 12-digit number hx1, x2, . . . ,...

A Universal Product Code (UPC) is a numerical representation of the bar codes used in stores, with an error-detecting feature to handle misscanned codes. A UPC is a 12-digit number hx1, x2, . . . , x12i where [∑ 6 i=1 3x2i−1 + x2i ] mod 10 = 0. (That is, the even-indexed digits plus three times the odd-indexed digits should be divisible by 10.) Prove that there exists a bijection between the set of 11-digit numbers and the set of valid 12-digit UPC codes. Use this fact to determine the number of valid UPC codes