It is related to a type of mathematics game

Question

Start with two digits from $1\dots9$ to form a $2$-digit number.

Multiply this number by a single digit from $1\dots9$.

Add the result to another two digit number from $1\dots9$, and calculate the result.

You can only use each digit once.

In other words:

How can the digits $\, 1,2,3,4,5,6,7,8,9 \,$ all be assigned to the placeholders $\, a,b,c,d,e,f,g,h,i \,$ so that $\, ab \times c = fg\,$ and $\, fg + de = hi \,$?

^{($ab$, $de$, $fg$ and $hi$ are explicit two-digit numbers, not products.)}

Answer 1

Assuming I understand what you are asking, the answer is:

$17 \times 4 \rightarrow 68 + 25 \rightarrow 93$

Answer 2

Assuming you mean that both the intermediate result ($ab*c$) and the result ($ab*c+de$) are included in the usage of digits condition.

Then

the only answer is $17\times4=68\rightarrow68+25=93$

---

Assuming you mean that the result ($ab*c+de$) is included in the usage of digits condition.

Then

there are $447$ valid answers, including $3$ which only use $[1,7]$:\begin{align}14\times2+37&=65\\14\times3+25&=67\\16\times2+43&=75\end{align}

Because

These are all of them:

12 * 3 + 49 =  85
12 * 3 + 58 =  94
12 * 4 + 37 =  85
12 * 4 + 39 =  87
13 * 2 + 49 =  75
13 * 2 + 68 =  94
13 * 4 + 26 =  78
13 * 5 + 24 =  89
31 * 6 + 89 = 275
31 * 7 + 48 = 265
31 * 8 + 47 = 295
14 * 2 + 37 =  65
14 * 2 + 39 =  67
14 * 2 + 65 =  93
14 * 2 + 59 =  87
14 * 2 + 67 =  95
14 * 3 + 25 =  67
14 * 3 + 27 =  69
14 * 3 + 56 =  98
41 * 5 + 68 = 273
41 * 6 + 39 = 285
41 * 6 + 79 = 325
41 * 7 + 82 = 369
41 * 7 + 95 = 382
41 * 8 + 29 = 357
41 * 8 + 67 = 395
15 * 3 + 24 =  69
15 * 3 + 42 =  87
15 * 3 + 29 =  74
15 * 3 + 47 =  92
51 * 4 + 69 = 273
51 * 4 + 79 = 283
51 * 7 + 82 = 439
51 * 7 + 29 = 386
51 * 8 + 29 = 437
51 * 9 + 27 = 486
16 * 2 + 43 =  75
16 * 2 + 57 =  89
16 * 3 + 47 =  95
61 * 3 + 95 = 278
16 * 4 + 23 =  87
16 * 4 + 25 =  89
61 * 4 + 35 = 279
61 * 4 + 53 = 297
61 * 4 + 85 = 329
61 * 5 + 87 = 392
61 * 5 + 79 = 384
61 * 7 + 32 = 459
61 * 8 + 39 = 527
61 * 8 + 49 = 537
61 * 9 + 24 = 573
17 * 2 + 34 =  68
17 * 2 + 35 =  69
17 * 2 + 64 =  98
17 * 2 + 49 =  83
17 * 3 + 45 =  96
71 * 3 + 46 = 259
17 * 4 + 25 =  93
17 * 4 + 28 =  96
71 * 4 + 85 = 369
71 * 4 + 68 = 352
71 * 5 + 29 = 384
71 * 5 + 68 = 423
71 * 6 + 32 = 458
71 * 6 + 98 = 524
71 * 8 + 26 = 594
71 * 9 + 43 = 682
18 * 2 + 43 =  79
18 * 2 + 39 =  75
18 * 2 + 57 =  93
18 * 3 + 42 =  96
18 * 3 + 25 =  79
81 * 3 + 54 = 297
18 * 4 + 23 =  95
18 * 4 + 25 =  97
81 * 4 + 52 = 376
81 * 4 + 72 = 396
81 * 6 + 43 = 529
81 * 7 + 26 = 593
81 * 9 + 24 = 753
81 * 9 + 35 = 764
18 * 9 + 74 = 236
19 * 2 + 36 =  74
19 * 2 + 38 =  76
19 * 2 + 45 =  83
19 * 2 + 47 =  85
19 * 3 + 27 =  84
91 * 5 + 23 = 478
91 * 5 + 32 = 487
91 * 6 + 32 = 578
91 * 6 + 28 = 574
91 * 7 + 45 = 682
91 * 8 + 26 = 754
91 * 8 + 34 = 762
23 * 1 + 45 =  68
32 * 1 + 54 =  86
23 * 1 + 64 =  87
32 * 1 + 46 =  78
23 * 1 + 56 =  79
32 * 1 + 65 =  97
23 * 1 + 75 =  98
32 * 1 + 57 =  89
23 * 4 + 95 = 187
32 * 4 + 59 = 187
23 * 4 + 67 = 159
32 * 4 + 67 = 195
23 * 5 + 64 = 179
23 * 5 + 74 = 189
23 * 5 + 69 = 184
23 * 6 + 47 = 185
23 * 6 + 49 = 187
24 * 1 + 63 =  87
42 * 1 + 36 =  78
24 * 1 + 65 =  89
42 * 1 + 56 =  98
24 * 1 + 59 =  83
24 * 3 + 15 =  87
24 * 3 + 17 =  89
24 * 3 + 95 = 167
24 * 3 + 87 = 159
24 * 6 + 35 = 179
24 * 6 + 53 = 197
42 * 7 + 91 = 385
42 * 8 + 61 = 397
25 * 1 + 43 =  68
52 * 1 + 34 =  86
25 * 1 + 73 =  98
52 * 1 + 37 =  89
25 * 1 + 39 =  64
25 * 1 + 64 =  89
52 * 1 + 46 =  98
25 * 1 + 48 =  73
25 * 1 + 68 =  93
25 * 3 + 14 =  89
25 * 3 + 89 = 164
52 * 8 + 63 = 479
26 * 1 + 53 =  79
62 * 1 + 35 =  97
26 * 1 + 49 =  75
26 * 1 + 57 =  83
26 * 3 + 17 =  95
26 * 4 + 79 = 183
62 * 7 + 19 = 453
62 * 8 + 41 = 537
27 * 1 + 38 =  65
27 * 1 + 56 =  83
27 * 1 + 59 =  86
27 * 1 + 68 =  95
27 * 3 + 14 =  95
27 * 3 + 15 =  96
27 * 3 + 84 = 165
27 * 3 + 68 = 149
72 * 4 + 81 = 369
27 * 4 + 85 = 193
27 * 5 + 34 = 169
27 * 5 + 63 = 198
72 * 6 + 51 = 483
72 * 8 + 43 = 619
72 * 8 + 39 = 615
28 * 1 + 37 =  65
28 * 1 + 39 =  67
28 * 1 + 45 =  73
28 * 1 + 65 =  93
28 * 1 + 67 =  95
28 * 3 + 65 = 149
82 * 4 + 51 = 379
28 * 4 + 63 = 175
28 * 4 + 57 = 169
82 * 4 + 67 = 395
82 * 6 + 47 = 539
82 * 7 + 45 = 619
82 * 9 + 16 = 754
28 * 9 + 65 = 317
29 * 1 + 35 =  64
29 * 1 + 38 =  67
29 * 1 + 54 =  83
29 * 1 + 46 =  75
29 * 1 + 57 =  86
29 * 3 + 67 = 154
29 * 3 + 78 = 165
92 * 4 + 17 = 385
29 * 4 + 67 = 183
92 * 7 + 14 = 658
92 * 7 + 41 = 685
34 * 1 + 52 =  86
43 * 1 + 25 =  68
34 * 1 + 58 =  92
34 * 2 + 17 =  85
34 * 2 + 19 =  87
34 * 2 + 97 = 165
43 * 2 + 79 = 165
34 * 2 + 89 = 157
43 * 2 + 89 = 175
43 * 5 + 71 = 286
43 * 5 + 81 = 296
43 * 5 + 76 = 291
34 * 6 + 87 = 291
34 * 7 + 51 = 289
34 * 7 + 18 = 256
34 * 7 + 58 = 296
35 * 1 + 62 =  97
53 * 1 + 26 =  79
35 * 1 + 29 =  64
35 * 1 + 47 =  82
53 * 2 + 68 = 174
53 * 4 + 69 = 281
35 * 7 + 41 = 286
35 * 7 + 19 = 264
53 * 7 + 91 = 462
35 * 7 + 46 = 291
53 * 8 + 72 = 496
53 * 8 + 67 = 491
36 * 1 + 42 =  78
63 * 1 + 24 =  87
36 * 1 + 49 =  85
36 * 1 + 58 =  94
36 * 2 + 15 =  87
36 * 2 + 17 =  89
63 * 2 + 49 = 175
36 * 2 + 87 = 159
36 * 4 + 75 = 219
63 * 5 + 97 = 412
63 * 7 + 51 = 492
63 * 7 + 18 = 459
63 * 9 + 17 = 584
37 * 1 + 52 =  89
73 * 1 + 25 =  98
37 * 1 + 28 =  65
37 * 1 + 45 =  82
37 * 1 + 49 =  86
37 * 2 + 15 =  89
37 * 2 + 94 = 168
37 * 5 + 84 = 269
37 * 5 + 96 = 281
73 * 6 + 21 = 459
73 * 6 + 51 = 489
37 * 6 + 59 = 281
73 * 8 + 12 = 596
73 * 8 + 41 = 625
73 * 8 + 45 = 629
73 * 9 + 24 = 681
37 * 9 + 82 = 415
38 * 1 + 27 =  65
38 * 1 + 29 =  67
38 * 1 + 54 =  92
38 * 1 + 56 =  94
38 * 2 + 69 = 145
38 * 4 + 65 = 217
38 * 4 + 67 = 219
83 * 5 + 76 = 491
38 * 6 + 51 = 279
38 * 6 + 17 = 245
38 * 6 + 19 = 247
83 * 9 + 15 = 762
39 * 1 + 25 =  64
39 * 1 + 28 =  67
39 * 1 + 46 =  85
39 * 1 + 47 =  86
39 * 2 + 67 = 145
39 * 2 + 76 = 154
39 * 2 + 78 = 156
39 * 2 + 87 = 165
93 * 5 + 21 = 486
93 * 5 + 17 = 482
39 * 5 + 81 = 276
39 * 6 + 41 = 275
93 * 6 + 14 = 572
39 * 6 + 47 = 281
93 * 8 + 12 = 756
93 * 8 + 21 = 765
45 * 1 + 23 =  68
54 * 1 + 32 =  86
45 * 1 + 28 =  73
54 * 1 + 29 =  83
45 * 1 + 37 =  82
54 * 1 + 38 =  92
45 * 3 + 62 = 197
54 * 3 + 27 = 189
45 * 7 + 81 = 396
54 * 7 + 18 = 396
46 * 1 + 32 =  78
64 * 1 + 23 =  87
46 * 1 + 52 =  98
64 * 1 + 25 =  89
46 * 1 + 29 =  75
46 * 1 + 39 =  85
46 * 2 + 83 = 175
46 * 2 + 93 = 185
46 * 2 + 95 = 187
64 * 2 + 59 = 187
64 * 3 + 95 = 287
64 * 7 + 81 = 529
46 * 7 + 29 = 351
46 * 7 + 59 = 381
64 * 8 + 17 = 529
46 * 8 + 27 = 395
64 * 8 + 27 = 539
47 * 1 + 35 =  82
47 * 1 + 39 =  86
47 * 3 + 28 = 169
74 * 3 + 59 = 281
47 * 5 + 91 = 326
47 * 5 + 63 = 298
47 * 5 + 86 = 321
47 * 6 + 13 = 295
47 * 8 + 16 = 392
74 * 8 + 23 = 615
48 * 1 + 25 =  73
48 * 2 + 63 = 159
48 * 2 + 73 = 169
84 * 3 + 15 = 267
84 * 3 + 17 = 269
48 * 3 + 25 = 169
48 * 3 + 52 = 196
48 * 3 + 75 = 219
48 * 6 + 71 = 359
84 * 6 + 19 = 523
48 * 6 + 27 = 315
84 * 6 + 27 = 531
48 * 6 + 29 = 317
84 * 7 + 51 = 639
48 * 7 + 16 = 352
48 * 7 + 25 = 361
84 * 7 + 25 = 613
48 * 7 + 29 = 365
48 * 7 + 56 = 392
49 * 1 + 26 =  75
49 * 1 + 36 =  85
49 * 1 + 37 =  86
49 * 3 + 28 = 175
49 * 3 + 68 = 215
49 * 5 + 31 = 276
49 * 5 + 18 = 263
49 * 5 + 81 = 326
49 * 5 + 36 = 281
49 * 5 + 67 = 312
49 * 5 + 76 = 321
49 * 6 + 81 = 375
94 * 6 + 23 = 587
94 * 7 + 23 = 681
49 * 7 + 25 = 368
94 * 7 + 25 = 683
94 * 8 + 13 = 765
56 * 1 + 23 =  79
65 * 1 + 32 =  97
56 * 1 + 42 =  98
65 * 1 + 24 =  89
56 * 1 + 27 =  83
65 * 1 + 28 =  93
56 * 1 + 38 =  94
56 * 2 + 37 = 149
56 * 3 + 81 = 249
65 * 3 + 84 = 279
56 * 3 + 49 = 217
56 * 4 + 97 = 321
56 * 7 + 91 = 483
56 * 8 + 31 = 479
56 * 8 + 23 = 471
57 * 1 + 32 =  89
75 * 1 + 23 =  98
57 * 1 + 26 =  83
57 * 1 + 29 =  86
57 * 2 + 49 = 163
57 * 2 + 69 = 183
75 * 3 + 64 = 289
57 * 3 + 48 = 219
57 * 4 + 61 = 289
57 * 4 + 63 = 291
57 * 4 + 98 = 326
57 * 6 + 81 = 423
57 * 6 + 89 = 431
57 * 8 + 13 = 469
57 * 8 + 36 = 492
58 * 1 + 34 =  92
58 * 1 + 36 =  94
58 * 2 + 63 = 179
58 * 2 + 47 = 163
85 * 3 + 14 = 269
85 * 3 + 41 = 296
85 * 3 + 19 = 274
58 * 3 + 67 = 241
58 * 4 + 61 = 293
58 * 4 + 37 = 269
58 * 4 + 39 = 271
58 * 6 + 73 = 421
85 * 7 + 24 = 619
85 * 7 + 34 = 629
59 * 1 + 24 =  83
59 * 1 + 27 =  86
59 * 3 + 71 = 248
59 * 3 + 84 = 261
59 * 3 + 87 = 264
59 * 4 + 31 = 267
59 * 4 + 76 = 312
59 * 6 + 18 = 372
59 * 6 + 24 = 378
59 * 6 + 27 = 381
59 * 6 + 78 = 432
59 * 7 + 13 = 426
67 * 1 + 28 =  95
76 * 2 + 43 = 195
67 * 2 + 49 = 183
67 * 4 + 15 = 283
67 * 5 + 93 = 428
67 * 8 + 13 = 549
68 * 1 + 25 =  93
68 * 1 + 27 =  95
68 * 2 + 43 = 179
68 * 2 + 39 = 175
68 * 2 + 57 = 193
68 * 3 + 47 = 251
68 * 4 + 57 = 329
68 * 4 + 79 = 351
68 * 7 + 43 = 519
69 * 2 + 45 = 183
69 * 2 + 47 = 185
69 * 3 + 74 = 281
69 * 4 + 51 = 327
69 * 4 + 81 = 357
69 * 5 + 42 = 387
69 * 5 + 73 = 418
69 * 5 + 78 = 423
69 * 5 + 87 = 432
69 * 7 + 38 = 521
69 * 7 + 48 = 531
69 * 8 + 21 = 573
78 * 3 + 15 = 249
78 * 3 + 61 = 295
87 * 4 + 21 = 369
87 * 6 + 12 = 534
87 * 6 + 21 = 543
78 * 6 + 23 = 491
78 * 6 + 25 = 493
79 * 4 + 52 = 368
79 * 4 + 65 = 381
79 * 5 + 31 = 426
79 * 5 + 23 = 418
79 * 6 + 38 = 512
79 * 8 + 21 = 653
79 * 8 + 13 = 645
89 * 4 + 16 = 372
89 * 5 + 31 = 476
89 * 5 + 17 = 462
89 * 5 + 26 = 471
89 * 6 + 13 = 547
89 * 7 + 12 = 635
89 * 7 + 31 = 654