help Counting

Hello Guest,

Of the three-digit integers greater than $500$, how many have two digits that are equal to each other and the remaining digit different from the other two

500, 511, 522, 533, 544, 566, 577, 588, 599 (10)

600, 611, 622, 633, 644, 655, 677, 688, 699 (10)

700, 711, 722, 733 ,744 ,755 ,766, 788, 799 (10)

800, 811, 822, 833, 844, 855, 866, 877, 899 (10)

900, 911, 922, 933, 944, 955, 966, 977, 988 (10)

505, 515, 525, 535, 545, 565, 575, 585, 595 (10)

606, 616, 626, 636, 646, 656, 676, 686, 696 (10)

707, 717, 727, 737, 747, 757, 767, 787, 797 (10)

808, 818, 828, 838, 848, 858, 868, 878, 898 (10)

909, 919, 929, 939, 949, 959, 969, 979, 989 (10)

550, 551, 552, 553, 554, 556, 557, 558, 559 (10)

660, 661, 662, 663, 664, 665, 667, 668, 669 (10)

770, 771, 772, 773, 774, 775, 776, 778, 779 (19)

880, 881, 882, 883, 884, 885, 886, 887, 889 (10)

This gives 140 numbers, if it were so desired.

Hope that was helpful. ^^

Straight

There are:

(500, 505, 511, 515, 522, 525, 533, 535, 544, 545, 550, 551, 552, 553, 554, 556, 557, 558, 559, 565, 566, 575, 577, 585, 588, 595, 599, 600, 606, 611, 616, 622, 626, 633, 636, 644, 646, 655, 656, 660, 661, 662, 663, 664, 665, 667, 668, 669, 676, 677, 686, 688, 696, 699, 700, 707, 711, 717, 722, 727, 733, 737, 744, 747, 755, 757, 766, 767, 770, 771, 772, 773, 774, 775, 776, 778, 779, 787, 788, 797, 799, 800, 808, 811, 818, 822, 828, 833, 838, 844, 848, 855, 858, 866, 868, 877, 878, 880, 881, 882, 883, 884, 885, 886, 887, 889, 898, 899, 900, 909, 911, 919, 922, 929, 933, 939, 944, 949, 955, 959, 966, 969, 977, 979, 988, 989, 990, 991, 992, 993, 994, 995, 996, 997, 998)==135 such integers.

Of the three-digit integers greater than $500$, how many have two digits that are equal to each other and the remaining digit different from the other two

2 zeros :  4

1 zero : 5*2=10

the rest have no zeros

2 of 1,2,3,4      :    4*5 = 20

2 of 5,6,7,8,9   and one 1,2,3 or 4  :  5*4*2=40

2 of 5,6,7,8,9   and one 5,6,7,8,9  :  5*4*3=60

4+10+20+40+60 = 134