To solve the problem of how many tickets Pentabhai purchased for his 200 friends, while avoiding ticket numbers that contain the digit '5', we can follow these steps:
### Step-by-Step Solution:
1. **Identify the Total Number of Tickets**:
Pentabhai needs to buy tickets for 200 friends, which means he needs to purchase 200 tickets.
2. **Determine the Range of Ticket Numbers**:
The ticket numbers start from 1 and go up to some number \( n \) that we need to find, where he can buy 200 tickets excluding those that contain the digit '5'.
3. **Count the Excluded Tickets**:
We need to find out how many ticket numbers from 1 to \( n \) contain the digit '5'.
4. **List of Excluded Tickets**:
The ticket numbers that contain '5' are:
- Single digit: 5
- Two digits: 15, 25, 35, 45, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59
- Three digits: 105, 115, 125, 135, 145, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 205, 215, 225, 235, 245, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 305, 315, 325, 335, 345, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 405, 415, 425, 435, 445, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 505, 515, 525, 535, 545, 550, 551, 552, 553, 554, 555, 556, 557, 558, 559, 605, 615, 625, 635, 645, 650, 651, 652, 653, 654, 655, 656, 657, 658, 659, 705, 715, 725, 735, 745, 750, 751, 752, 753, 754, 755, 756, 757, 758, 759, 805, 815, 825, 835, 845, 850, 851, 852, 853, 854, 855, 856, 857, 858, 859, 905, 915, 925, 935, 945, 950, 951, 952, 953, 954, 955, 956, 957, 958, 959, 1005, 1015, 1025, 1035, 1045, 1050, 1051, 1052, 1053, 1054, 1055, 1056, 1057, 1058, 1059, 1105, 1115, 1125, 1135, 1145, 1150, 1151, 1152, 1153, 1154, 1155, 1156, 1157, 1158, 1159, 1205, 1215, 1225, 1235, 1245, 1250, 1251, 1252, 1253, 1254, 1255, 1256, 1257, 1258, 1259, 1305, 1315, 1325, 1335, 1345, 1350, 1351, 1352, 1353, 1354, 1355, 1356, 1357, 1358, 1359, 1405, 1415, 1425, 1435, 1445, 1450, 1451, 1452, 1453, 1454, 1455, 1456, 1457, 1458, 1459, 1505, 1515, 1525, 1535, 1545, 1550, 1551, 1552, 1553, 1554, 1555, 1556, 1557, 1558, 1559, 1605, 1615, 1625, 1635, 1645, 1650, 1651, 1652, 1653, 1654, 1655, 1656, 1657, 1658, 1659, 1705, 1715, 1725, 1735, 1745, 1750, 1751, 1752, 1753, 1754, 1755, 1756, 1757, 1758, 1759, 1805, 1815, 1825, 1835, 1845, 1850, 1851, 1852, 1853, 1854, 1855, 1856, 1857, 1858, 1859, 1905, 1915, 1925, 1935, 1945, 1950, 1951, 1952, 1953, 1954, 1955, 1956, 1957, 1958, 1959, 2005, 2015, 2025, 2035, 2045, 2050, 2051, 2052, 2053, 2054, 2055, 2056, 2057, 2058, 2059, 2105, 2115, 2125, 2135, 2145, 2150, 2151, 2152, 2153, 2154, 2155, 2156, 2157, 2158, 2159, 2205, 2215, 2225, 2235, 2245, 2250, 2251, 2252, 2253, 2254, 2255, 2256, 2257, 2258, 2259, 2305, 2315, 2325, 2335, 2345, 2350, 2351, 2352, 2353, 2354, 2355, 2356, 2357, 2358, 2359, 2405, 2415, 2425, 2435, 2445, 2450, 2451, 2452, 2453, 2454, 2455, 2456, 2457, 2458, 2459, 2505, 2515, 2525, 2535, 2545, 2550, 2551, 2552, 2553, 2554, 2555, 2556, 2557, 2558, 2559, 2605, 2615, 2625, 2635, 2645, 2650, 2651, 2652, 2653, 2654, 2655, 2656, 2657, 2658, 2659, 2705, 2715, 2725, 2735, 2745, 2750, 2751, 2752, 2753, 2754, 2755, 2756, 2757, 2758, 2759, 2805, 2815, 2825, 2835, 2845, 2850, 2851, 2852, 2853, 2854, 2855, 2856, 2857, 2858, 2859, 2905, 2915, 2925, 2935, 2945, 2950, 2951, 2952, 2953, 2954, 2955, 2956, 2957, 2958, 2959, 3005, 3015, 3025, 3035, 3045, 3050, 3051, 3052, 3053, 3054, 3055, 3056, 3057, 3058, 3059, 3105, 3115, 3125, 3135, 3145, 3150, 3151, 3152, 3153, 3154, 3155, 3156, 3157, 3158, 3159, 3205, 3215, 3225, 3235, 3245, 3250, 3251, 3252, 3253, 3254, 3255, 3256, 3257, 3258, 3259, 3305, 3315, 3325, 3335, 3345, 3350, 3351, 3352, 3353, 3354, 3355, 3356, 3357, 3358, 3359, 3405, 3415, 3425, 3435, 3445, 3450, 3451, 3452, 3453, 3454, 3455, 3456, 3457, 3458, 3459, 3505, 3515, 3525, 3535, 3545, 3550, 3551, 3552, 3553, 3554, 3555, 3556, 3557, 3558, 3559, 3605, 3615, 3625, 3635, 3645, 3650, 3651, 3652, 3653, 3654, 3655, 3656, 3657, 3658, 3659, 3705, 3715, 3725, 3735, 3745, 3750, 3751, 3752, 3753, 3754, 3755, 3756, 3757, 3758, 3759, 3805, 3815, 3825, 3835, 3845, 3850, 3851, 3852, 3853, 3854, 3855, 3856, 3857, 3858, 3859, 3905, 3915, 3925, 3935, 3945, 3950, 3951, 3952, 3953, 3954, 3955, 3956, 3957, 3958, 3959, 4005, 4015, 4025, 4035, 4045, 4050, 4051, 4052, 4053, 4054, 4055, 4056, 4057, 4058, 4059, 4105, 4115, 4125, 4135, 4145, 4150, 4151, 4152, 4153, 4154, 4155, 4156, 4157, 4158, 4159, 4205, 4215, 4225, 4235, 4245, 4250, 4251, 4252, 4253, 4254, 4255, 4256, 4257, 4258, 4259, 4305, 4315, 4325, 4335, 4345, 4350, 4351, 4352, 4353, 4354, 4355, 4356, 4357, 4358, 4359, 4405, 4415, 4425, 4435, 4445, 4450, 4451, 4452, 4453, 4454, 4455, 4456, 4457, 4458, 4459, 4505, 4515, 4525, 4535, 4545, 4550, 4551, 4552, 4553, 4554, 4555, 4556, 4557, 4558, 4559, 4605, 4615, 4625, 4635, 4645, 4650, 4651, 4652, 4653, 4654, 4655, 4656, 4657, 4658, 4659, 4705, 4715, 4725, 4735, 4745, 4750, 4751, 4752, 4753, 4754, 4755, 4756, 4757, 4758, 4759, 4805, 4815, 4825, 4835, 4845, 4850, 4851, 4852, 4853, 4854, 4855, 4856, 4857, 4858, 4859, 4905, 4915, 4925, 4935, 4945, 4950, 4951, 4952, 4953, 4954, 4955, 4956, 4957, 4958, 4959, 5005, 5015, 5025, 5035, 5045, 5050, 5051, 5052, 5053, 5054, 5055, 5056, 5057, 5058, 5059, 5105, 5115, 5125, 5135, 5145, 5150, 5151, 5152, 5153, 5154, 5155, 5156, 5157, 5158, 5159, 5205, 5215, 5225, 5235, 5245, 5250, 5251, 5252, 5253, 5254, 5255, 5256, 5257, 5258, 5259, 5305, 5315, 5325, 5335, 5345, 5350, 5351, 5352, 5353, 5354, 5355, 5356, 5357, 5358, 5359, 5405, 5415, 5425, 5435, 5445, 5450, 5451, 5452, 5453, 5454, 5455, 5456, 5457, 5458, 5459, 5505, 5515, 5525, 5535, 5545, 5550, 5551, 5552, 5553, 5554, 5555, 5556, 5557, 5558, 5559, 5605, 5615, 5625, 5635, 5645, 5650, 5651, 5652, 5653, 5654, 5655, 5656, 5657, 5658, 5659, 5705, 5715, 5725, 5735, 5745, 5750, 5751, 5752, 5753, 5754, 5755, 5756, 5757, 5758, 5759, 5805, 5815, 5825, 5835, 5845, 5850, 5851, 5852, 5853, 5854, 5855, 5856, 5857, 5858, 5859, 5905, 5915, 5925, 5935, 5945, 5950, 5951, 5952, 5953, 5954, 5955, 5956, 5957, 5958, 5959, 6005, 6015, 6025, 6035, 6045, 6050, 6051, 6052, 6053, 6054, 6055, 6056, 6057, 6058, 6059, 6105, 6115, 6125, 6135, 6145, 6150, 6151, 6152, 6153, 6154, 6155, 6156, 6157, 6158, 6159, 6205, 6215, 6225, 6235, 6245, 6250, 6251, 6252, 6253, 6254, 6255, 6256, 6257, 6258, 6259, 6305, 6315, 6325, 6335, 6345, 6350, 6351, 6352, 6353, 6354, 6355, 6356, 6357, 6358, 6359, 6405, 6415, 6425, 6435, 6445, 6450, 6451, 6452, 6453, 6454, 6455, 6456, 6457, 6458, 6459, 6505, 6515, 6525, 6535, 6545, 6550, 6551, 6552, 6553, 6554, 6555, 6556, 6557, 6558, 6559, 6605, 6615, 6625, 6635, 6645, 6650, 6651, 6652, 6653, 6654, 6655, 6656, 6657, 6658, 6659, 6705, 6715, 6725, 6735, 6745, 6750, 6751, 6752, 6753, 6754, 6755, 6756, 6757, 6758, 6759, 6805, 6815, 6825, 6835, 6845, 6850, 6851, 6852, 6853, 6854, 6855, 6856, 6857, 6858, 6859, 6905, 6915, 6925, 6935, 6945, 6950, 6951, 6952, 6953, 6954, 6955, 6956, 6957, 6958, 6959, 7005, 7015, 7025, 7035, 7045, 7050, 7051, 7052, 7053, 7054, 7055, 7056, 7057, 7058, 7059, 7105, 7115, 7125, 7135, 7145, 7150, 7151, 7152, 7153, 7154, 7155, 7156, 7157, 7158, 7159, 7205, 7215, 7225, 7235, 7245, 7250, 7251, 7252, 7253, 7254, 7255, 7256, 7257, 7258, 7259, 7305, 7315, 7325, 7335, 7345, 7350, 7351, 7352, 7353, 7354, 7355, 7356, 7357, 7358, 7359, 7405, 7415, 7425, 7435, 7445, 7450, 7451, 7452, 7453, 7454, 7455, 7456, 7457, 7458, 7459, 7505, 7515, 7525, 7535, 7545, 7550, 7551, 7552, 7553, 7554, 7555, 7556, 7557, 7558, 7559, 7605, 7615, 7625, 7635, 7645, 7650, 7651, 7652, 7653, 7654, 7655, 7656, 7657, 7658, 7659, 7705, 7715, 7725, 7735, 7745, 7750, 7751, 7752, 7753, 7754, 7755, 7756, 7757, 7758, 7759, 7805, 7815, 7825, 7835, 7845, 7850, 7851, 7852, 7853, 7854, 7855, 7856, 7857, 7858, 7859, 7905, 7915, 7925, 7935, 7945, 7950, 7951, 7952, 7953, 7954, 7955, 7956, 7957, 7958, 7959, 8005, 8015, 8025, 8035, 8045, 8050, 8051, 8052, 8053, 8054, 8055, 8056, 8057, 8058, 8059, 8105, 8115, 8125, 8135, 8145, 8150, 8151, 8152, 8153, 8154, 8155, 8156, 8157, 8158, 8159, 8205, 8215, 8225, 8235, 8245, 8250, 8251, 8252, 8253, 8254, 8255, 8256, 8257, 8258, 8259, 8305, 8315, 8325, 8335, 8345, 8350, 8351, 8352, 8353, 8354, 8355, 8356, 8357, 8358, 8359, 8405, 8415, 8425, 8435, 8445, 8450, 8451, 8452, 8453, 8454, 8455, 8456, 8457, 8458, 8459, 8505, 8515, 8525, 8535, 8545, 8550, 8551, 8552, 8553, 8554, 8555, 8556, 8557, 8558, 8559, 8605, 8615, 8625, 8635, 8645, 8650, 8651, 8652, 8653, 8654, 8655, 8656, 8657, 8658, 8659, 8705, 8715, 8725, 8735, 8745, 8750, 8751, 8752, 8753, 8754, 8755, 8756, 8757, 8758, 8759, 8805, 8815, 8825, 8835, 8845, 8850, 8851, 8852, 8853, 8854, 8855, 8856, 8857, 8858, 8859, 8905, 8915, 8925, 8935, 8945, 8950, 8951, 8952, 8953, 8954, 8955, 8956, 8957, 8958, 8959, 9005, 9015, 9025, 9035, 9045, 9050, 9051, 9052, 9053, 9054, 9055, 9056, 9057, 9058, 9059, 9105, 9115, 9125, 9135, 9145, 9150, 9151, 9152, 9153, 9154, 9155, 9156, 9157, 9158, 9159, 9205, 9215, 9225, 9235, 9245, 9250, 9251, 9252, 9253, 9254, 9255, 9256, 9257, 9258, 9259, 9305, 9315, 9325, 9335, 9345, 9350, 9351, 9352, 9353, 9354, 9355, 9356, 9357, 9358, 9359, 9405, 9415, 9425, 9435, 9445, 9450, 9451, 9452, 9453, 9454, 9455, 9456, 9457, 9458, 9459, 9505, 9515, 9525, 9535, 9545, 9550, 9551, 9552, 9553, 9554, 9555, 9556, 9557, 9558, 9559, 9605, 9615, 9625, 9635, 9645, 9650, 9651, 9652, 9653, 9654, 9655, 9656, 9657, 9658, 9659, 9705, 9715, 9725, 9735, 9745, 9750, 9751, 9752, 9753, 9754, 9755, 9756, 9757, 9758, 9759, 9805, 9815, 9825, 9835, 9845, 9850, 9851, 9852, 9853, 9854, 9855, 9856, 9857, 9858, 9859, 9905, 9915, 9925, 9935, 9945, 9950, 9951, 9952, 9953, 9954, 9955, 9956, 9957, 9958, 9959.
5. **Count the Total Excluded Tickets**:
From the above list, we can count the total number of tickets that contain the digit '5' from 1 to 200. These are:
- 5, 15, 25, 35, 45, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59 (15 tickets)
- 105, 115, 125, 135, 145, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159 (15 tickets)
- 205, 215, 225, 235, 245, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259 (15 tickets)
- Total excluded tickets = 38
6. **Calculate the Total Tickets Purchased**:
Since Pentabhai needs to buy 200 tickets and 38 of them contain the digit '5', he will need to buy tickets up to \( 200 + 38 = 238 \).
7. **Final Answer**:
The last ticket number that Pentabhai purchased is **238**.
