just see this approach comment if anything wrong i did)
total edges=9 for mst number of edges must be in graph=v-1=5
step 1 : total combination
9c5=126
but this combination contains 3 edges,4edges,5 edges cycle.so subtraction from these must be done.
step 2: finding 3 edges cycle
3 edges cycle are (162,652,523,534)- 4 cycles
now for each cycle 2 edges are remaining so choosing those 2 edges will take 6c2 ways.
total=4*6c2
but if we choose 2 more edges then there may be chances of getting 4 edges cycle.
for example- in cycle 162 if we choose edge 6-5 and 2-5 cycle 1652 is possible,in cycle 625 if we choose 1-6 1-2 then 1652 is possible also in same cycle if we choose edge 5-3 2-3 then cycle 6532 is possible.
3 EDGES CYCLE EDGES ADDED 4 EDGES CYCLE FORMED
162 6-5 & 2-5 1652
625 1-6,1-2 OR 2-3,5-3 1652 &6532
235 6-5,2-6 OR 5-4,3-4 6532,5234
534 2-5,2-3 5234
so total 6 4-edges cycle will be formed so we must remove this 6 cycles
4*6c2-6=54
case 3: finding 4 edges cycle (1652,6253,2345)
choosing the fifth edges will take 5c1 ways but we have to careful that choosing 5th edges can lead us to 5 edges cycle.
4 EDGES CYCLE EDGES ADDED 5 EDGES CYCLE
1652 6-2 265216
6523 2-5 253265
2345 5-3 532543
So 3*5c1-3=12
case 4:choosing 5 edges cycle(123561,623456,253265,621652,352345)
total 5
final answer:126-(54+12+5)=55
check this out .comment if anything wrong