The number of different ways in which five alike dashes and eight alike dots can be arranged, using only seven of these ‘dashes’ and ‘dots’ is equal to

Question

The number of different ways in which five alike dashes and eight alike dots can be arranged, using only seven of these ‘dashes’ and ‘dots’ is equal to (A) 1287 (B) 119 (C) 120 (D) 1235520

Tardigrade Admin · Accepted Answer

Dashes 5 Dots 8 Arrangements 5 2 7 C2 4 3 7 C3 3 4 7 C4 2 5 7 C5 1 6 7 C6 0 7 7 C7 Required number of ways = 7 C2+ 7 C3+ 7 C4+ 7 C5+ 7 C6+ 7 C7 =27-7-1=120

Dashes 5	Dots 8	Arrangements
5	2	$^{7} C_{2}$
4	3	$^{7} C_{3}$
3	4	$^{7} C_{4}$
2	5	$^{7} C_{5}$
1	6	$^{7} C_{6}$
0	7	$^{7} C_{7}$

Q. The number of different ways in which five alike dashes and eight alike dots can be arranged, using only seven of these ‘dashes’ and ‘dots’ is equal to

1287

119

120

1235520

Dashes 5 Dots 8 Arrangements

5 2 $^{7} C_{2}$

4 3 $^{7} C_{3}$

3 4 $^{7} C_{4}$

2 5 $^{7} C_{5}$

1 6 $^{7} C_{6}$

0 7 $^{7} C_{7}$

Required number of ways $=^{7} C_{2} +^{7} C_{3} +^{7} C_{4} +^{7} C_{5} +^{7} C_{6} +^{7} C_{7}$
$= 2^{7} - 7 - 1 = 120$

Q. The number of different ways in which five alike dashes and eight alike dots can be arranged, using only seven of these ‘dashes’ and ‘dots’ is equal to

1287

119

120

1235520

Dashes 5 Dots 8 Arrangements 5 2 7C2​ 4 3 7C3​ 3 4 7C4​ 2 5 7C5​ 1 6 7C6​ 0 7 7C7​ Required number of ways =7C2​+7C3​+7C4​+7C5​+7C6​+7C7​ =27−7−1=120

Dashes 5 Dots 8 Arrangements

5 2 $^{7} C_{2}$

4 3 $^{7} C_{3}$

3 4 $^{7} C_{4}$

2 5 $^{7} C_{5}$

1 6 $^{7} C_{6}$

0 7 $^{7} C_{7}$

Required number of ways $=^{7} C_{2} +^{7} C_{3} +^{7} C_{4} +^{7} C_{5} +^{7} C_{6} +^{7} C_{7}$
$= 2^{7} - 7 - 1 = 120$